Answer:
C) Yes, because the joint probability of P and M is equal to 0.
Step-by-step explanation:
Given that, P represents the event that the selected person preferred pepperoni only.
And M represents the event that the selected person preferred mushroom only.
so, P and M are two mutually exclusive events because two events are mutually exclusive if they do not occur together.
Here, P and M cannot occur at same time.
so, the joint probability of P and M i.e) probability of both P and M to occur at same time is zero.
In the given problem we have 55 percent preferred pepperoni only, 30 percent preferred mushroom only, and 15 percent preferred something other than pepperoni and mushroom. The probability of event can be identified by selecting how many people can chose pepperoni and mushroom.
The correct option is (c)
Given:
Pepperoni preferred by 55 percent.
Mushroom preferred by 30 percent.
The people who preferred other than Mushroom and Pepperoni are 15 percent.
In the given problem, A person can not select both pepperoni and mushroom because percentages of the pizza preferences adds to 1 in this survey. So, the mathematically,
P(P∩M)=0P(P∩M)=0
The above expression indicates the events are mutually exclusive.
Therefore, the joint probability of P and M is equal to 0.
Thus, the correct option is (c)
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solve triangle ABC
c = 10, B = 35°, C = 65°
Question options: A = 80°, a = 10, b = 6.3
A = 80°, a = 6.3, b = 10.9
A = 80°, a = 10.9, b = 6.3
A = 80°, a = 73.6, b = 6.3
That figure obviously doesn't go with this problem. It doesn't matter; this is triangle ABC labeled the usual way.
c = 10, B = 35°, C = 65°
We have two angles and a side. The third angle is obviously
A = 180° - 35°- 65° = 80°
The remaining sides are given by the Law of Sines,
[tex]\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]
[tex]a = \dfrac{c \sin A}{\sin C} = \dfrac{10 \sin 80}{\sin 65} = 10.866[/tex]
[tex]b = \dfrac{c \sin B}{\sin C} = \dfrac{10 \sin 35}{\sin 65} = 6.328[/tex]
Answer: A=80°, a=10.9, b=6.3, third choice
Suppose we are testing people to see if the rate of use of seat belts has changed from a previous value of 88%. Suppose that in our random sample of 500 people, we see that 450 have the seat belt fastened. (a) About how many of 500 would we expect to be using their seat belts if the proportion who use seat belts is unchanged?
(b) We observe 450 people out of a random sample of 500 using their seatbelt. The p-value is 0.167. Explain the meaning of the p-value.
Answer:
a) We would expect to see 500*0.88=440
b) [tex]z=\frac{0.9 -0.88}{\sqrt{\frac{0.88(1-0.88)}{500}}}=1.376[/tex]
[tex]p_v =2*P(Z>1.376)=0.167[/tex]
So the p value obtained was a very high value and using the significance level assumed [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the true proportion is not significant different from 0.9.
The p value is a criterion to decide if we reject or not the null hypothesis, when [tex]p_v <\alpha[/tex] we reject the null hypothesis in other case we FAIL to reject the null hypothesis. And represent the "probability of obtaining the observed results of a test, assuming that the null hypothesis is correct".
Step-by-step explanation:
Data given and notation
n=500 represent the random sample taken
X=450 represent the people that have the seat belt fastened
[tex]\hat p=\frac{450}{500}=0.9[/tex] estimated proportion of people that have the seat belt fastened
[tex]p_o=0.88[/tex] is the value that we want to test
[tex]\alpha[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v{/tex} represent the p value (variable of interest)
Part a
We would expect to see 500*0.88=440
Part b
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the true proportion changes fro m 0.88.:
Null hypothesis:[tex]p=0.88[/tex]
Alternative hypothesis:[tex]p \neq 0.88[/tex]
When we conduct a proportion test we need to use the z statisitc, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.9 -0.88}{\sqrt{\frac{0.88(1-0.88)}{500}}}=1.376[/tex]
Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level assumed is [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a bilateral test the p value would be:
[tex]p_v =2*P(Z>1.376)=0.167[/tex]
So the p value obtained was a very high value and using the significance level assumed [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the true proportion is not significant different from 0.9.
The p value is a criterion to decide if we reject or not the null hypothesis, when [tex]p_v <\alpha[/tex] we reject the null hypothesis in other case we FAIL to reject the null hypothesis. And represent the "probability of obtaining the observed results of a test, assuming that the null hypothesis is correct".
A vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, blueberry, and pistachio ice cream, served on either a waffle, sugar, or plain cone. How many different single-scoop ice-cream cones can you buy from this vendor?
Answer:
The different single-scoop ice-cream cones can you buy from this vendor=12
Step-by-step explanation:
Given that a vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, blueberry, and pistachio ice cream, served on either a waffle, sugar, or plain cone.
Different types of icecreams are 4
Types of cones are 3
For each ice cream say we can have either waffle, sugar or plain cone.
i.e. for each type of ice cream, we have 3 different cones.
Hence for 4 types of ice creams we have 3*4=12 different single scoop ice creamcones.
There are 15 different single-scoop ice-cream cones that can be bought from the vendor.
Explanation:To calculate the number of different single-scoop ice-cream cones that can be bought from the vendor, we need to multiply the number of ice cream flavors with the number of cone types. The vendor offers 5 flavors and 3 cone types, so the total number of different single-scoop ice-cream cones that can be bought is 5 x 3 = 15.
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Drawing a 2, then drawing another 2?
The probability of drawing a 2 and then drawing another 2 in the board game is 1/72.
To calculate the probability of drawing a 2 and then drawing another 2 in the board game, we first need to understand the concept of probability. Probability represents the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). In this case, we're interested in the probability of two specific events happening sequentially: first drawing a 2, and then drawing another 2.
In the board game scenario, there are 9 numbers in total. So, the probability of drawing a 2 on the first draw is 1 out of 9 since there's one 2 among the nine numbers. After drawing the first number and not replacing it, there are now 8 numbers left. If the first draw was a 2, then there's one less 2 available among the remaining 8 numbers. So, the probability of drawing another 2 on the second draw is 1 out of 8.
To find the overall probability of both events happening in sequence (drawing a 2 and then drawing another 2), we multiply the probabilities of each event. Thus, the probability of drawing a 2 and then drawing another 2 is (1/9) * (1/8) = 1/72.
Therefore, the probability of drawing a 2 and then drawing another 2 in the board game is indeed 1/72. This means that out of all the possible sequences of two draws without replacement, only 1 out of 72 sequences will result in drawing a 2 followed by another 2.
Complete question :- In a board game, students draw a number do not replace it and then draw a second number
Determine the probability of each event occurring,
1
6
6
9
21
62
What is the probability of drawing a 2, then drawing another 2?
Alexis runs a small business and creates recordings of her friend’s skateboard stunts. She puts the recordings onto blu-ray and sells them online. She realized that the average cost of each blu-ray depends on the number of blu-rays she creates, because of the fixed cost. The cost of producing x blu-rays is given by
B(x) = 1250 +1.75x
1) Alexis wants to figure out the price to charge friends for the blu-rays. She doesn’t want to make money at this point since it is a brand new business, but does want to cover her costs. Suppose Alexis created 50 blu-rays. What is the cost of producing those 50 blu-rays? How much is it for each blu-ray?
please help my algabra grade dpends on it
Answer:
The cost of producing those 50 blu-rays is given by, 1337.5 unit and it is 26.75 unit for each blu-ray.
Step-by-step explanation:
The cost of producing x blu-rays is given by,
B(x) = 1250 + 1.75x unit -----------------------------(1)
So, cost of producing 50 blu-rays is given by,
B(50) = [tex]1250 + 50 \times (1.75)[/tex] unit
= (1250 + 87.5) unit
= 1337.5 unit
So, for each blu-ray, it is,
[tex]\frac {1337.5}{50}[/tex] unit
= 26.75 unit
The TreeDropp’d Fruit Company wants to sell its apples overseas in attractive pink and yellow gift boxes. To design the boxes, the company needs to estimate the average diameter of their apples. A random sample of 50 apples has a mean of 4.1 inches. You assume a population standard deviatino of .4 inches.
A. What is the point estimate in this scenario? Is it an unbiased estimator?
B. What is the critical z-value for a 95% interval?
C. What is the margin of error for a 95% interval in this scenario? Show your work.
Answer:
a)For this case the best estimator for the true mean is the sample mean [tex]\hat \mu =\bar X[/tex] because:
[tex]E(\bar X)= \frac{\sum_{i=1}^n E(X_i)}{n}[/tex]
And if we assume that each observation [tex]X_1 , X_2,...,X_n [/tex] follows a normal distribution [tex]X_i \sim N(\mu,\sigma)[/tex] then we have:
[tex]E(\bar X)=\frac{1}{n} n\mu =\mu[/tex]
So then yes the [tex]\bar X[/tex[ is a unbiased estimator for the true mean.
b) [tex]z_{\alpha/2}=1.96[/tex]
c) [tex] ME= 1.96 \frac{0.4}{\sqrt{50}}=0.1109[/tex]
Step-by-step explanation:
Previous concepts
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
[tex]\bar X=4.1[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
[tex]\sigma=0.4[/tex] represent the population standard deviation
n=50 represent the sample size
Assuming the X follows a normal distribution
[tex]X \sim N(\mu, \sigma=0.4[/tex]
The sample mean [tex]\bar X[/tex] is distributed on this way:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
A. What is the point estimate in this scenario? Is it an unbiased estimator?
For this case the best estimator for the true mean is the sample mean [tex]\hat \mu =\bar X[/tex] because:
[tex]E(\bar X)= \frac{\sum_{i=1}^n E(X_i)}{n}[/tex]
And if we assume that each observation [tex]X_1 , X_2,...,X_n [/tex] follows a normal distribution [tex]X_i \sim N(\mu,\sigma)[/tex] then we have:
[tex]E(\bar X)=\frac{1}{n} n\mu =\mu[/tex]
So then yes the [tex]\bar X[/tex[ is a unbiased estimator for the true mean.
B. What is the critical z-value for a 95% interval?
The next step would be find the value of [tex]\z_{\alpha/2}[/tex], [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex]
Using the normal standard table, excel or a calculator we see that:
[tex]z_{\alpha/2}=1.96[/tex]
C. What is the margin of error for a 95% interval in this scenario? Show your work.
The margin of error is given by:
[tex] ME= z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]
If we replace the values that we have we got:
[tex] ME= 1.96 \frac{0.4}{\sqrt{50}}=0.1109[/tex]
The confidence interval on this case is given by:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since we have all the values we can replace:
[tex]4.1 - 1.96\frac{0.4}{\sqrt{50}}=3.989[/tex]
[tex]4.1 + 1.96\frac{0.4}{\sqrt{50}}=4.211[/tex]
So on this case the 95% confidence interval would be given by (3.989;4.211)
For each situation, state the null and alternative hypotheses: (Type "mu" for the symbol μ , e.g. mu > 1 for the mean is greater than 1, mu < 1 for the mean is less than 1, mu not = 1 for the mean is not equal to 1. Please do not include units such as "mm" or "$" in your answer.)
a) The diameter of a spindle in a small motor is supposed to be 2.5 millimeters (mm) with a standard deviation of 0.17 mm. If the spindle is either too small or too large, the motor will not work properly. The manufacturer measures the diameter in a sample of 17 spindles to determine whether the mean diameter has moved away from the required measurement. Suppose the sample has an average diameter of 2.57 mm.
H0:
Ha:
(b) Harry thinks that prices in Caldwell are lower than the rest of the country. He reads that the nationwide average price of a certain brand of laundry detergent is $16.35 with standard deviation $2.20. He takes a sample from 3 local Caldwell stores and finds the average price for this same brand of detergent is $14.40.
H0:
Ha:
Answer:
Part (A) [tex]H_0:\mu=2.5[/tex] and [tex]H_a:\mu\neq2.5[/tex]
Part (B) [tex]H_0:\mu=16.35[/tex] and [tex]H_a:\mu<16.35[/tex]
Step-by-step explanation:
Consider the provided information.
Null hypothesis refers the population parameter is equal to the claimed value.
If there is no statistical significance in the test then it is know as the null which is denoted by [tex]H_0[/tex], otherwise it is known as alternative hypothesis denoted by [tex]H_a[/tex].
Part (A): The diameter of a spindle in a small motor is supposed to be 2.5 millimeters (mm) if the spindle is either too small or too large, the motor will not work properly.
Thus, the required null and alternative hypothesis are:
[tex]H_0:\mu=2.5[/tex] and [tex]H_a:\mu\neq2.5[/tex]
Part (B) The nationwide average price of a certain brand of laundry detergent is $16.35 and Harry thinks that prices in Caldwell are lower than the rest of the country.
Thus, the required null and alternative hypothesis are:
[tex]H_0:\mu=16.35[/tex] and [tex]H_a:\mu<16.35[/tex]
The null and alternative hypotheses for the given situations are as follows: a) H0: μ = 2.5 mm, Ha: μ ≠ 2.5 mm. b) H0: μ = $16.35, Ha: μ < $16.35.
Explanation:a) H0: The mean diameter of the spindles is 2.5 mm (μ = 2.5 mm).
Ha: The mean diameter of the spindles is not equal to 2.5 mm (μ ≠ 2.5 mm).
b) H0: The mean price of the laundry detergent in Caldwell is $16.35 (μ = $16.35).
Ha: The mean price of the laundry detergent in Caldwell is less than $16.35 (μ < $16.35).
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A recent study by a major financial investment company was interested in determining whether the annual percentage change in stock price for companies is linearly related with the annual percent change in profits for the company. The following data was determined for 7 randomly selected companies: Col1 % Change Stock Price (Y) 8.4 9.5 13.6 -3.2 7 18.4 -2.1Col2 % Change in Profit (X) 4.2 5.6 11.2 4.5 12.2 12 -13.4 Based upon this sample information, which of the following is the regression equation? a. y? 1.19-3.00 x b· y?=-4.198-0.612x c. y? =+4.198 x d, y? = 4.198 + 0.612 x
Answer:
c. y? =+4.198 x
Step-by-step explanation:
Hello!
Using the given data you need to estimate the equation of linear regression.
Dependent variable:
Y: Annual percentage change in stock price for a company.
Independent variable:
X: Annual percentage change in profits for the company.
The population regression line equation is:
[tex]Y_i= \alpha + \beta X_i + E_i[/tex]
To estimate the equation you need to find the point estimator for α and β.
The following formulas are the ones to use:
α ⇒ a= y[bar] - bX[bar]
β ⇒ b= (∑xy - [(∑x)(∑y)]/n)/(∑X²-(∑x)²/n)
As you can see you need to make several summatories before calculating the values of a and b:
n= 7
∑x= 36.30
∑x²= 667.09
∑y= 51.60
∑y²= 747.98
∑xy= 560.74
Sample mean of de dependent variable Y[bar]= ∑y/n= (51.60/7)= 7.37
Sample mean of the independent variable X[bar]= ∑x/n= (36.30/7)= 5.19
b= [tex]\frac{560.74-\frac{(36.3*51.6)}{7} }{667.09-\frac{(51.60)^2}{7} }[/tex]
b= 0.6122
a= [tex]7.37-(0.61*5.19)[/tex]
a= 4.196
The estimated regression equation is:
Y= 4.196 + 0.6122x
I hope it helps!
An article in an educational journal discussing a non-conventional high school reading program reports a 95% confidence interval of (520 , 560) for the average score on the reading portion of the SAT. The study was based on a random sample of 20 students and it was assumed that SAT scores would be approximately normally distributed.
Determine the margin of error associated with this confidence interval.
A. 20
B. 10
C. 15
D. 40
E. 5
Answer:
Option A is right
Step-by-step explanation:
Given that an article in an educational journal discussing a non-conventional high school reading program reports a 95% confidence interval of (520 , 560) for the average score on the reading portion of the SAT.
Let X be the score on the reading portion of the SAT.
By central limit theorem x bar, the sample mean will follow a normal.
95% confidence interval shows that mean
= average of the two limits
= 540
Margin of error = upper bound of confidence interval - Mean
=[tex]560-540=20[/tex]
Option A is right
A certain human red blood cell has a diameter of 0.000007 meters. Which expression represents this diameter, in meters, in scientific notation?
A) 7* 10^-6
B) 7* 10^-5
C) 7* 10^6
D) 7* 10^5
47.5% of children say that chocolate chip cookie is their favorite kind of cookie. If you randomly select 4 children find the probability that: a) exactly 2 say that chocolate chip cookie is their favorite. c) at least one of the 4 say that chocolate chip cookie is their favorite. d) Find the mean and standard deviation for this sample.
Answer:
a) 0.3731
b) 0.9241
c) Mean = 1.9
Standard Deviation = 0.9987
Step-by-step explanation:
We are given the following information:
P(chocolate chip cookie) = 47.5% = 0.475
Then the number of children follows a binomial distribution, where
[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]
where n is the total number of observations, x is the number of success, p is the probability of success.
Now, we are given n = 4
a) x = 2
We have to evaluate:
[tex]P(x =2)\\= \binom{4}{2}(0.475)^2(1-0.475)^{4-2}\\= 0.3731[/tex]
b) x = 1
We have to evaluate
[tex]P(x \geq 1) = 1 - P(x = 0)\\= 1 - \binom{4}{0}(0.475)^0(1-0.475)^{4-0}\\= 1 - 0.0759\\= 0.9241[/tex]
c) Mean and standard deviation of distribution
[tex]\text{Mean} = np = 4\times 0.475 = 1.9\\\text{Standard deviation} = \sqrt{np(1-p)} = \sqrt{4\times 0.475(1-0.475)} = 0.9987[/tex]
In a study of automobile collision insurance costs, a random sample of n = 35 repair costs of front-end damage caused by hitting a wall at a specified speed had a mean of $1,438. (a) Given that σ = $269 for such data, what can be said with 98% confidence about the maximum error if x = $1, 438 is used as an estimate of the average cost of such repairs.
Answer: The maximum error = $105.76.
Step-by-step explanation:
Formula to find the maximum error:
[tex]E= z^*\dfrac{\sigma}{\sqrt{n}}[/tex]
, where n= sample size.
[tex]\sigma[/tex] = Population standard deviation
z*= Critical value(two-tailed).
As per given , we have
[tex]\overline{x}=1438[/tex]
n= 35
[tex]\sigma=269[/tex]
For 98% confidence , the significance level = [tex]1-0.98=0.02[/tex]
By z-table , the critical value (two -tailed) =[tex]z^* = z_{\alpha/2}=z_{0.01}=2.326[/tex]
Now , the maximum error = [tex]E= (2.326)\dfrac{269}{\sqrt{35}}[/tex]
[tex]E= (2.326)\dfrac{269}{5.9160797831}[/tex]
[tex]E= (2.326)\times45.4692989044=105.761589252\pprox105.76[/tex]
Hence, With 98% confidence level , the maximum error = $105.76.
A market research group is interested in comparing the mean weight loss for two different popular diets. The researcher chooses two random samples of participants for the two diet programs. For Diet A, the mean weekly weight loss for 10 participants was 1.5 pounds with standard deviation 0.4 pounds. For Diet B, the mean weekly weight loss for 12 participants was 1.2 pounds with standard deviation 0.6 pounds. At a 5% significance level, does this indicate that Diet A is better than Diet B?
Answer:
ok
Step-by-step explanation:Jawo
A market research group is interested in comparing the mean weight loss for two different popular diets. And this indicates that Diet A is better than Diet B.
What is hypothesis?An assumption or concept is given as a hypothesis for the purpose of debating it and testing if it might be true.
Given:
For diet A :
Let n₁ = 10 and X₁ = 1.5, σ₁ = 0.4
For diet B:
Let n₂ = 12 and X₂ = 1.2, σ₂ = 0.6
And α = 0.05
Hypothesis:
H₀: μ₁ = μ₂
H₁ : μ₁ > μ₂
The test:
t = (x₁ - x₂) / S√(1/n₁ + 1/n₂)
Where S = √(n₁σ₁² + n₂σ₂²)/(n₁+n₂-2)
S = √(1.6+4.32)/(20)
S= √(0.296)
S = 0.544
Now substituting the value of S to the test.
t = (x₁ - x₂) / S√(1/n₁ + 1/n₂)
t = (1.5-1.2)/ (0.544)√(1/10 + 1/12)
t = (0.3) / (0.544)(0.428)
t = (0.3) / (0.2329)
t = 1.29
So the p-value is 0.212463888
Here, α = 0.05 > 0.212463888
If P value < α then reject H₀ at % level of significance, accept otherwise.
H₁ : μ₁ > μ₂
So, Diet A is better than Diet B
Therefore, diet A is better than diet B
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A guidance counselor at a university is investigating demand for study abroad. The question before her is how engineering and humanities majors compare regarding interest in study abroad during summer. Random samples of 20 engineering and humanities majors each were interviewed. Eight engineering majors and 12 humanities majors expressed interest in study abroad during the summer. The estimated difference in the proportion of engineering and humanities majors, p E − p H , where pE is the proportion of engineering majors interested in study abroad and pH is the proportion for humanities, has sampling distribution:a. with mean μ=0.2 and standard deviation σ=0.1549 shape not known.b. Normal with mean μ= -0.2 and standard deviation σ=0.1549.c. Normal with mean μ= 0.2 and standard deviation σ=0.1549. d. with mean μ=-0.2 and standard deviation σ=0.1549 shape not known
Answer:
[tex]\mu_{p_E -p_H} = 0.4-0.6 =-0.2[/tex]
[tex]SE_{p_E -p_H}=\sqrt{\frac{0.4(1-0.4)}{20} +\frac{0.6 (1-0.6)}{20}}=0.1549[/tex]
b. Normal with mean μ= -0.2 and standard deviation σ=0.1549
Step-by-step explanation:
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]p_E[/tex] represent the real population proportion for Engineering
[tex]\hat p_E =\frac{8}{20}=0.4[/tex] represent the estimated proportion for Engineering
[tex]n_E=20[/tex] is the sample size required for Engineering
[tex]p_H[/tex] represent the real population proportion for Humanities
[tex]\hat p_H =\frac{12}{20}=0.6[/tex] represent the estimated proportion for Humanities
[tex]n_H=20[/tex] is the sample size required for Humanities
We assume that the population proportions follows a normal distribution since we satisfy these conditions:
[tex]np\geq 5[/tex] ,[tex]n(1-p)\geq 5[/tex]
[tex]20*0.4= 8\geq 5 , 20(1-0.4)=12\geq 5[/tex]
The population proportion have the following distribution
[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]
And we are interested on the distribution for [tex]p_E-p_H[/tex]
For this case we know that the distribution for the differences of proportions is also normal and given by:
[tex]p_E -p_H \sim N(\hat p_E -\hat p_H, \sqrt{\frac{\hat p_E(1-\hat p_E)}{n_E} +\frac{\hat p_H (1-\hat p_H)}{n_H}}[/tex]
So then we can find the mean and the deviation like this:
[tex]\mu_{p_E -p_H} = 0.4-0.6 =-0.2[/tex]
[tex]SE_{p_E -p_H}=\sqrt{\frac{0.4(1-0.4)}{20} +\frac{0.6 (1-0.6)}{20}}=0.1549[/tex]
So then the best option is :
b. Normal with mean μ= -0.2 and standard deviation σ=0.1549
The shape is unknown since there is no knowledge about it and the representative sample is limited.
Given that;Total number of interview in engineering and humanities = 20
Number of interview in engineering = 8
Number of interview in humanities = 12
Computation:⇒ Probability of interview in engineering - Probability of interview in humanities
⇒ 8/20 - 12/20
⇒ -0.20
The shape is unknown since there is no knowledge about it and the representative sample is limited.
Therefore, Option D is the right answer.
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Consider the following hypothesis test:H 0: = 17H a: 17A sample of 40 provided a sample mean of 14.12. The population standard deviation is 4.a. Compute the value of the test statistic (to 2 decimals). (If answer is negative, use minus "-" sign.)b. What is the p-value (to 4 decimals)?c. Using = .05, can it be concluded that the population mean is not equal to 17? SelectYesNoItem 3Answer the next three questions using the critical value approach.d. Using = .05, what are the critical values for the test statistic (to 2 decimals)? ±e. State the rejection rule: Reject H 0 if z is Selectgreater than or equal togreater thanless than or equal toless thanequal tonot equal toItem 5 the lower critical value and is Selectgreater than or equal togreater thanless than or equal toless thanequal tonot equal toItem 6 the upper critical value.f. Can it be concluded that the population mean is not equal to 17?
Answer:
We conclude that the population mean is not equal to 17.
Step-by-step explanation:
We are given the following in the question:
Population mean, μ = 17
Sample mean, [tex]\bar{x}[/tex] = 14.12
Sample size, n = 40
Alpha, α = 0.05
Population standard deviation, σ = 4
First, we design the null and the alternate hypothesis
[tex]H_{0}: \mu = 17\\H_A: \mu \neq 17[/tex]
We use Two-tailed z test to perform this hypothesis.
a) Formula:
[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]
Putting all the values, we have
[tex]z_{stat} = \displaystyle\frac{14.12 - 17}{\frac{4}{\sqrt{40}} } = -4.5536[/tex]
b) P-value can be calculated from the standard z-table.
P-value = 0.0000
c) Since the p-value is less than the significance level, we reject the null hypothesis and accept the alternate hypothesis. Thus, the population mean is not equal to 17
d) Now, [tex]z_{critical} \text{ at 0.05 level of significance } = \pm 1.96[/tex]
e) Rejection Rule:
We reject the null hypothesis if it is less than lower critical value and greater than the upper critical value
If the z-statistic lies outside the acceptance region which is from -1.96 to +1.96, we reject the null hypothesis.
f) Since the calculated z-stat lies outside the acceptance region, we reject the null hypothesis and accept the alternate hypothesis. Thus, the population mean is not equal to 17.
The test statistic is -1.78 and the p-value is 0.0761, indicating that we fail to reject the null hypothesis. Therefore, it cannot be concluded that the population mean is not equal to 17.
Explanation:The test statistic can be calculated using the formula:
test statistic = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
Plugging in the given values, we get:
test statistic = (14.12 - 17) / (4 / sqrt(40))
Calculating this gives us a test statistic value of -1.78.
The p-value can be calculated using the test statistic. We need to find the probability that a test statistic at least as extreme as -1.78 would occur assuming the null hypothesis is true. Using a standard normal distribution table or software, we find the p-value to be approximately 0.0761.
Since the p-value is greater than the significance level (alpha = 0.05), we fail to reject the null hypothesis. Therefore, we can conclude that there is not enough evidence to suggest that the population mean is not equal to 17.
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The television show Degenerate Housewives has been successful for many years. That show recently had a share of 18, meaning that among the TV sets in use, 18% were tuned to Degenerate Housewives. Assume that an advertiser wants to verify that 18% share value by conducting its own survey, and a pilot survey begins with 12 households have TV sets in use at the time of a Degenerate Housewivesbroadcast.Find the probability that none of the households are tuned to Degenerate Housewives. P(none) =____________ Find the probability that at least one household is tuned to Degenerate Housewives. P(at least one) = __________Find the probability that at most one household is tuned to Degenerate Housewives. P(at most one) = ___________If at most one household is tuned to Degenerate Housewives, does it appear that the 18% share value is wrong? (Hint: Is the occurrence of at most one household tuned to Degenerate Housewives unusual?)
Answer:
Step-by-step explanation:
Given that the television show Degenerate Housewives has been successful for many years. That show recently had a share of 18, meaning that among the TV sets in use, 18% were tuned to Degenerate Housewives.
Each household is independent of the other and there two outcomes.
X no of households that are tuned to Degenerate Housewives. is Bin (12, 0.18)
the probability that none of the households are tuned to Degenerate Housewives. P(none) =_______0.0924_____
the probability that at least one household is tuned to Degenerate Housewives. P(at least one) = ____1-0.0924 = 0.9076______
the probability that at most one household is tuned to Degenerate Housewives. P(at most one) = _____0.3359______
If at most one household is tuned to Degenerate Housewives, does it appear that the 18% share value is wrong? (Hint: Is the occurrence of at most one household tuned to Degenerate Housewives unusual?)
No 33% is not unusual.
Use the inner product〈f,g〉=∫10f(x)g(x)dxin the vector space C0[0,1] of continuous functions on the domain [0,1] to find 〈f,g〉, ∥f∥, ∥g∥, and the angle αf,g between f(x) and g(x) forf(x)=−10x2−6 and g(x)=−9x−4.〈f,g〉= ,∥f∥= ,∥g∥= ,αf,g .
The answer to this problem involves applying integrals, norms, and concepts of angles between vectors to the functions f(x) and g(x). The INNER PRODUCT is the integral of the products of the two functions, the norms are the square roots of the inner products of the functions with themselves, and the angle between the functions is calculated using the dot product and norms.
Explanation:To find the inner product 〈f,g〉, the norms ∥f∥ and ∥g∥, and the angle αf,g between the functions f(x)=−10x2−6 and g(x)=−9x−4, we'll apply concepts from vector calculus. The inner product (also known as the dot product) is the integral from 0 to 1 of the products of the two functions. The norm of a function is the square root of the inner product of the function with itself. The angle between two vectors in a Vector Space, in this case the space of continuous functions C0[0,1], is given by cos(α) = 〈f,g〉/( ∥f∥∙ ∥g∥). Integrating and solving these equations will give us the desired values.
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These are the corrected values for the inner product, norms, and angle between [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]in the vector space [tex]\(C^0[0,1]\).[/tex]
1. The inner product \(\langle f,g \rangle\) is given by the integral [tex]\(\int_{1}^{0} f(x)g(x) \, dx\).[/tex] We will compute this integral with[tex]\(f(x) = -10x^2 - 6\) and \(g(x) = -9x - 4\).[/tex]
[tex]\[ \langle f,g \rangle = \int_{1}^{0} (-10x^2 - 6)(-9x - 4) \, dx \][/tex]
Expanding the integrand, we get:
[tex]\[ \int_{1}^{0} (90x^3 + 40x^2 + 54x + 24) \, dx \][/tex]
Integrating term by term, we have:
[tex]\[ \left[ \frac{90}{4}x^4 + \frac{40}{3}x^3 + \frac{54}{2}x^2 + 24x \right]_{1}^{0} \][/tex]
[tex]\[ = \left( \frac{90}{4} \cdot 0^4 + \frac{40}{3} \cdot 0^3 + \frac{54}{2} \cdot 0^2 + 24 \cdot 0 \right) - \left( \frac{90}{4} \cdot 1^4 + \frac{40}{3} \cdot 1^3 + \frac{54}{2} \cdot 1^2 + 24 \cdot 1 \right) \][/tex]
[tex]\[ = - \left( \frac{90}{4} + \frac{40}{3} + \frac{54}{2} + 24 \right) \][/tex]
[tex]\[ = - \left( 22.5 + 13.333 + 27 + 24 \right) \][/tex]
[tex]\[ = -87.833 \][/tex]
So, [tex]\(\langle f,g \rangle = -87.833\).[/tex]
2. The norm of f is given by [tex]\(\|f\|[/tex] = [tex]\sqrt{\langle f,f \rangle}\)[/tex], where:
[tex]\[ \langle f,f \rangle = \int_{1}^{0} (-10x^2 - 6)^2 \, dx \][/tex]
[tex]\[ = \int_{1}^{0} (100x^4 + 120x^2 + 36) \, dx \][/tex]
[tex]\[ = \left[ \frac{100}{5}x^5 + \frac{120}{3}x^3 + 36x \right]_{1}^{0} \][/tex]
[tex]\[ = - \left( \frac{100}{5} + \frac{120}{3} + 36 \right) \][/tex]
[tex]\[ = - \left( 20 + 40 + 36 \right) \][/tex]
[tex]\[ = -96 \][/tex]
[tex]\[ \|f\| = \sqrt{96} \][/tex]
3. Similarly, the norm of g is given by [tex]\(\|g\|[/tex] = [tex]\sqrt{\langle g,g \rangle}\)[/tex], where:
[tex]\[ \langle g,g \rangle = \int_{1}^{0} (-9x - 4)^2 \, dx \][/tex]
[tex]\[ = \int_{1}^{0} (81x^2 + 72x + 16) \, dx \][/tex]
[tex]\[ = \left[ \frac{81}{3}x^3 + 36x^2 + 16x \right]_{1}^{0} \][/tex]
[tex]\[ = - \left( \frac{81}{3} + 36 + 16 \right) \][/tex]
[tex]\[ = - \left( 27 + 36 + 16 \right) \][/tex]
[tex]\[ = -79 \][/tex]
[tex]\[ \|g\| = \sqrt{79} \][/tex]
4. The angle \[tex](\alpha_{f,g}\)[/tex] between [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] is found using the formula:
[tex]\[ \cos(\alpha_{f,g}) = \frac{\langle f,g \rangle}{\|f\| \cdot \|g\|} \][/tex]
[tex]\[ \alpha_{f,g} = \cos^{-1}\left(\frac{-87.833}{\sqrt{96} \cdot \sqrt{79}}\right) \][/tex]
[tex]\[ \langle f,g \rangle = -87.833, \quad \|f\| = \sqrt{96}, \quad \|g\| = \sqrt{79}, \quad \alpha_{f,g} = \cos^{-1}\left(\frac{-87.833}{\sqrt{96} \cdot \sqrt{79}}\right) \][/tex]
The correct calculations for the inner product and norms should be as follows:
[tex]\[ \langle f,g \rangle = \int_{0}^{1} (90x^3 + 40x^2 + 54x + 24) \, dx \][/tex]
[tex]\[ \|f\| = \sqrt{\int_{0}^{1} (100x^4 + 120x^2 + 36) \, dx} \][/tex]
[tex]\[ \|g\| = \sqrt{\int_{0}^{1} (81x^2 + 72x + 16) \, dx} \][/tex]
The angle calculation remains the same. Let's correct the integrals:
[tex]\[ \langle f,g \rangle = \left[ \frac{90}{4}x^4 + \frac{40}{3}x^3 + \frac{54}{2}x^2 + 24x \right]_{0}^{1} \][/tex]
[tex]\[ = \frac{90}{4} + \frac{40}{3} + \frac{54}{2} + 24 \][/tex]
[tex]\[ = \frac{90}{4} + \frac{40}{3} + \frac{54}{2} + 24 \][/tex]
[tex]\[ = 87.833 \][/tex]
So,[tex]\(\langle f,g \rangle = 87.833\).[/tex]
[tex]\[ \|f\| = \sqrt{\left[ \frac{100}{5}x^5 + \frac{120}{3}x^3 + 36x \right]_{0}^{1}} \][/tex]
[tex]\[ = \sqrt{20 + 40 + 36} \][/tex]
[tex]\[ = \sqrt{96} \][/tex]
[tex]\[ \|g\| = \sqrt{\left[ \frac{81}{3}x^3 + 36x^2 + 16x \right]_{0}^{1}} \][/tex]
[tex]\[ = \sqrt{27 + 36 + 16} \][/tex]
[tex]\[ = \sqrt{79} \][/tex]
The corrected values are:
[tex]\[ \langle f,g \rangle = 87.833, \quad \|f\| = \sqrt{96}, \quad \|g\| = \sqrt{79} \][/tex]
The angle calculation remains the same:
[tex]\[ \alpha_{f,g} = \cos^{-1}\left(\frac{87.833}{\sqrt{96} \cdot \sqrt{79}}\right) \][/tex]
Congress regulates corporate fuel economy and sets an annual gas mileage for cars. A company with a large fleet of cars hopes to meet the 2011 goal of 30.2mpg or better for their fleet of cars. To see if the goal is being met, they check the gasoline usage for 50 company trips chosen at random, finding a mean of 32.12mpg and a standard deviation of 4.83mpg. In this strong evidence that they have attained their fuel economy goal? a. Define the parameter and state the hypotheses. b. Define the sampling distribution (mean and standard deviation). c. Perform the test and calculate P-value d. State your conclusion. e. Explain what the p-value means in this context.
Answer:
a) Null hypothesis:[tex]\mu \leq 30.2[/tex]
Alternative hypothesis:[tex]\mu > 30.2[/tex]
b) [tex]X \sim N(\mu=32.12, \sigma=4.83)[/tex]
And the distribution for the random sample is given by:
[tex]\bar X \sim N(\mu=32.12, \frac{\sigma}{\sqrt{n}}=\frac{4.83}{\sqrt{50}}=0.683)[/tex]
c) [tex]t=\frac{32.12-30.2}{\frac{4.83}{\sqrt{50}}}=2.81[/tex]
[tex]p_v =P(t_{(49)}>2.81)=0.0035[/tex]
d) If we compare the p value and the significance level assumed [tex]\alpha=0.01[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is significantly higher than 30.2.
e) The p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct. And for this case is a value to accept or reject the null hypothesis.
Step-by-step explanation:
1) Data given and notation
[tex]\bar X=32.12[/tex] represent the sample mean
[tex]s=4.83[/tex] represent the sample standard deviation
[tex]n=50[/tex] sample size
[tex]\mu_o =30.2[/tex] represent the value that we want to test
[tex]\alpha[/tex] represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
a. Define the parameter and state the hypotheses.
We need to conduct a hypothesis in order to check if the mean is higher than 30.2 mpg, the system of hypothesis would be:
Null hypothesis:[tex]\mu \leq 30.2[/tex]
Alternative hypothesis:[tex]\mu > 30.2[/tex]
If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:
[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
b. Define the sampling distribution (mean and standard deviation).
Let X the random variable who represent the variable of interest. And we know that the distribution for X is:
[tex]X \sim N(\mu=32.12, \sigma=4.83)[/tex]
And the distribution for the random sample is given by:
[tex]\bar X \sim N(\mu=32.12, \frac{\sigma}{\sqrt{n}}=\frac{4.83}{\sqrt{50}}=0.683)[/tex]
c. Perform the test and calculate P-value
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]t=\frac{32.12-30.2}{\frac{4.83}{\sqrt{50}}}=2.81[/tex]
P-value
The first step is calculate the degrees of freedom, on this case:
[tex]df=n-1=50-1=49[/tex]
Since is a one right tailed test the p value would be:
[tex]p_v =P(t_{(49)}>2.81)=0.0035[/tex]
d. State your conclusion.
If we compare the p value and the significance level assumed [tex]\alpha=0.01[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is significantly higher than 30.2.
e. Explain what the p-value means in this context.
The p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct. And for this case is a value to accept or reject the null hypothesis.
Final answer:
Explanation of hypothesis testing for a fuel economy standard compliance claim.
Explanation:
Hypotheses:
Parameter: Average fuel economyNull Hypothesis (H0): μ = 35.5Alternative Hypothesis (Ha): μ < 35.5Sampling Distribution:
Mean: 34.6 mpgStandard Deviation: 10.3 mpgPerforming the Test:
Calculate the Z-score and P-value from the sample data using the formula. Compare the P-value to the significance level.
Conclusion: Since **p-value = 0.2578** is greater than the significance level of 0.05, we fail to reject the null hypothesis. There is not enough evidence to claim that the fleet meets the fuel economy standard.
A group of veterinary researchers plan a study to estimate the average number of enteroliths in horses suffering from them. The number of enteroliths is determined by surgery or ultrasound. This requires the participation of a radiologist and a surgeon in the study. The researchers decide that they should only enroll horses seen at the best veterinary hospital in the country to ensure correct counts. Such a hospital will typically see the most severe cases. The researchers will enroll the first 100 horses seen beginning on the day the study is funded. The resulting interval is likely biased because:
A. the best veterinary hospital is likely to see the most severe cases with larger number of enteroliths.
B. the first 100 horses seen is not an SRS of all horses with enteroliths and therefore never representative.
C. a sample of 100 horses is too small to obtain an unbiased estimate.
D. All of the above.
The sample of 100 horses is too small to obtain an unbiased estimate.
What is Applied research?Applied research frequently focuses more on finding solutions to particular issues that directly impact people in the present. As an illustration, a social psychologist conducting fundamental research on violence would consider the various potential causes of violence in general.
Given, A study is being planned by a group of veterinary researchers to determine how many enteroliths are present on average in affected horses. When the study is financed, the researchers will sign up the first 100 horses they observe.
A study plan, which is considered to be the most important piece of information describing the projected research sample of an investigator, is the main part of an application. An important analyst is also given the opportunity to speak on the planned study, highlighting its benefits and explaining the implementation and conduct of the research. Because it covers your whole research project from start to finish and identifies and explains your emphasis, technique, and goals, a research plan is crucial.
Therefore, The sample size of 100 horses is insufficient to provide a fair assessment.
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Wedding expenses and marriage length: With the headline, "Want a happy marriage? Have a big, cheap wedding" CNN reported on a study that examined the correlation between wedding expenses and the length of marriages. The news article states, "A new study found that couples who spend less on their wedding tend to have longer-lasting marriages than those who splurge.What would be a reasonable explanation for the observed correlation? A. Having an inexpensive wedding helps young couples avoid financial burdens that many strain their marriage. B. Having an inexpensive wedding has no impact on the length of marriage because the cost of a wedding is a contounding variable that explains the correlation C. Having an inexpensive wedding guarantees a couple will have a longmarriage the study shows a the term because strong correlation between two variables
Answer:
A
Step-by-step explanation:
Having an inexpensive wedding helps young couples avoid financial burdens that may strain their marriage. This is true as it will ensure the young couples spend wisely. They understand that if they splurge, the returns will not be as great as that from a long-lasting marriage.
The analyst in the Dorben Reference Library decides to use the work sampling technique to establish standards. Twenty employees are involved. The operations include cataloging, charging books out, returning books to their proper location, cleaning books, record keeping, packing books for shipment, and handling correspondence. A preliminary investigation resulted in the estimate that 30 percent of the time of the group was spent in cataloging. How many work sampling observations would be made if it were desirable to be 95 percent confident that the observed data were within a tolerance of ±10 percent of the population data? Describe how the random observations should be made.
Answer:
At least 81 observations should be made to be 95% confident that the observed data is within a tolerance of ±10 percent of the population data.
Step-by-step explanation:
The following equation is used to compute the minimum sample size required to estimate the population proportion within the required margin of error:
n≥ p×(1-p) × [tex](\frac{z}{ME} )^2[/tex] where
n is the sample sizep is the estimated proportion of the time of the group was spent in cataloging (30% or 0.30)z is the corresponding z-score for 95% confidence level (1.96) ME is the margin of error (tolerance) in the estimation (10% or 0.10)Then, n≥ 0.30×0.70 × [tex](\frac{1.96}{0.10} )^2[/tex] ≈ 80.67
At least 81 observations should be made to be 95% confident that the observed data is within a tolerance of ±10 percent of the population data.
Random observations should include different employees, and sampling time should also be random.
Find the volume of the region bounded by the paraboloid z=x2+y2 and the plane x+y=1 in the first octant.
Answer:
Step-by-step explanation:
Given
Paraboloid [tex]z=x^2+y^2[/tex] and Plane [tex]x+y=1[/tex] are bounded to get the area
In first octant
value of x varies from [tex]0\leq x\leq 1[/tex]
value of y varies from [tex]0\leq y\leq 1[/tex]
Volume [tex]V=\int_{0}^{1}\int_{0}^{1-x}\left ( x^2+y^2\right )dydx[/tex]
[tex]V=\int_{0}^{1}\left [ x^2y+\frac{y^3}{3}\right ]_{0}^{1-x}[/tex]
[tex]V=\int_{0}^{1}\left [ x^2(1-x)+\frac{(1-x)^3}{3}\right ]dx[/tex]
[tex]V=\int_{0}^{1}\left ( \frac{-4x^3}{3}+2x^2-x+\frac{1}{3}\right )dx[/tex]
[tex]V=\frac{1}{6}[/tex]
The volume of the region bounded by the paraboloid z=x2+y2 and the plane x+y=1 in the first octant is 1/6
The paraboloid and the plane are given as:
[tex]z=x^2+y^2[/tex]
[tex]x+y = 1[/tex]
Make y the subject
[tex]y = 1 - x[/tex]
The plane is in the first octant.
So, the volume is represented as:
[tex]V = \int\limits^1_0 { \int\limits^y_0 {z^2} \, dy } \, dx[/tex]
The integral becomes
[tex]V = \int\limits^1_0 { \int\limits^{1-x}_0 {x^2 + y^2} \, dy } \, dx[/tex]
Integrate with respect to y
[tex]V = \int\limits^1_0 [x^2y + \frac 13y^3]\limits^{1-x}_0 } \, dx[/tex]
Expand
[tex]V = \int\limits^1_0 [x^2(1-x) + \frac 13(1-x)^3]} \, dx[/tex]
Expand
[tex]V = \int\limits^1_0 [x^2-x^3 + \frac 13(1 -x + x^2 - x^3)]} \, dx[/tex]
Expand
[tex]V = \int\limits^1_0 [x^2-x^3 + \frac 13 -x + x^2 - \frac {x^3}3]} \, dx[/tex]
Evaluate the like terms
[tex]V = \int\limits^1_0 [ \frac 13 -x + 2x^2 - \frac {4x^3}3]} \, dx[/tex]
Integrate
[tex]V = [\frac 13x -\frac 12x^2 + \frac 23x^3 - \frac 13x^4]\limits^1_0[/tex]
Expand
[tex]V = \frac 13(1) -\frac 12(1)^2 + \frac 23(1)^3 - \frac 13(1)[/tex]
[tex]V = -\frac 12(1)^2 + \frac 23(1)^3[/tex]
Evaluate the exponents
[tex]V = -\frac 12 + \frac 23[/tex]
Add the fractions
[tex]V = \frac{-3 + 4}{6}[/tex]
[tex]V = \frac{1}{6}[/tex]
Hence, the volume of the region is 1/6
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An article in the Journal of Composite Materials (Vol 23, 1989, p. 1200) describes the effect of delamination on the natural frequency of beams made from composite laminates. Five such delaminated beams were subjected to loads, and the resulting frequencies were as follows (in Hz):
230.66, 233.05, 232.58, 229.48, 232.58
(a) Find a 90% two-sided CI on mean natural frequency. Round your answers to 2 decimal places..
(b) Do the results of your calculations support the claim that mean natural frequency is 235 Hz?
Answer:
a) The 90% confidence interval would be given by (230.212;233.128)
b) For this case if we analyze the confidence interval we see that not contains the value of 235. So we can't support the claim that the true mean is higher than 235 Hz at 10% of significance.
Step-by-step explanation:
Previous concepts and data given
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Data: 230.66, 233.05, 232.58, 229.48, 232.58
We can calculate the mean and the deviation from these data with the following formulas:
[tex]\bar X= \frac{\sum_{i=1}^n x_i}{n}[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}[/tex]
[tex]\bar X=231.67[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=1.531 represent the sample standard deviation
n=5 represent the sample size
Part a) Confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=5-1=4[/tex]
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,4)".And we see that [tex]t_{\alpha/2}=2.13[/tex]
Now we have everything in order to replace into formula (1):
[tex]231.67-2.13\frac{1.531}{\sqrt{5}}=230.212[/tex]
[tex]231.67+2.13\frac{1.531}{\sqrt{5}}=233.128[/tex]
So on this case the 90% confidence interval would be given by (230.212;233.128)
Part b) Do the results of your calculations support the claim that mean natural frequency is 235 Hz?
For this case if we analyze the confidence interval we see that not contains the value of 235. So we can't support the claim that the true mean is higher than 235 Hz at 10% of significance.
To determine the 90% confidence interval for mean natural frequency, calculate the mean, standard deviation, find the t-value, and compute the margin of error and CI. The claim that the mean frequency is 235 Hz is assessed by checking if this value is within the calculated CI.
Explanation:To find the 90% two-sided confidence interval (CI) on mean natural frequency for the five delaminated beams, we first calculate the mean (\( \bar{x} \)) and the standard deviation (s) of the given frequencies. Then, we use the t-distribution, as the sample size is small (n=5), to find the t-value for a 90% CI, which corresponds to a significance level (alpha) of 0.10 and degrees of freedom (df) of n-1. Finally, we apply the formula for the 90% CI as follows:
Calculate the mean: \( \bar{x} = (230.66 + 233.05 + 232.58 + 229.48 + 232.58) / 5 = 231.67 \) Hz.Calculate the standard deviation (s):(b) To assess whether the mean natural frequency is 235 Hz, we would check if this value lies within the 90% CI. If it does not, the results do not support the claim that the mean natural frequency is 235 Hz.
There are 9 observatories tracking a near-Earth asteroid, and each independently estimates how close it will come to Earth next year, in millions of miles, with the results. 481, .439,448,446,618, 411, 250, 604, 414. Previous experience suggests these are unbiased estimates of the true distance d, with a standard deviation of 15. a) We estimate the true mean of these distance estimates by d = i. b) Assuming this is a large enough sample, write down a 94% confidence interval for the true distance d.
Answer:
94% Confidence interval: (0.3771 ,0.5365)
Step-by-step explanation:
We are given the following data set:
0.481, 0.439,0.448,0.446,0.618, 0.411, 0.250, 0.604, 0.414
a) True mean of these distance estimates by d
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{4.111}{9} = 0.4568[/tex]
b)
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
Sum of squares of differences = 0.0954
[tex]S.D = \sqrt{\frac{0.0954}{8}} = 0.1092[/tex]
94% Confidence interval:
[tex]\bar{x} \pm t_{critical}\displaystyle\frac{s}{\sqrt{n}}[/tex]
Putting the values, we get,
[tex]t_{critical}\text{ at degree of freedom 8 and}~\alpha_{0.06} = \pm 2.1891[/tex]
[tex]0.4568 \pm 2.1891(\frac{0.1092}{\sqrt{9}} ) = 0.4568 \pm 0.0797 = (0.3771 ,0.5365)[/tex]
Final answer:
To calculate the 94% confidence interval for the true mean distance of an asteroid, one needs to compute the sample mean, standard error, and then apply the appropriate Z-score to determine the range.
Explanation:
The student's question is related to calculating a confidence interval for the true mean distance of a near-Earth asteroid. With the given unbiased estimates of the distances and a standard deviation of 15, we will calculate the 94% confidence interval for the true distance d.
Firstly, calculate the sample mean X by adding up all the given distance estimates and dividing by the number of observatories, which is 9.
Next, find the standard error (SE) of the mean by dividing the standard deviation by the square root of the number of observatories (n).
To determine the 94% confidence interval, we'll use the Z-score associated with it (approximately 1.88) and the standard error.
The confidence interval formula is X ± (Z * SE). We calculate the upper and lower bounds of the interval accordingly.
By following these steps, the student can establish the 94% confidence interval, giving them a range within which the true mean distance of the asteroid is likely to fall.
Suppose that a random sample of size 49 is to be selected from a population with mean 49 and standard deviation 6. What is the approximate probability that Xwll be more than 0.5 away from the population mean?
a 0.4403
b) 0.6403
c) 0.8807
d) 0.0664
e) 0.5597
f) None of the above
Answer:
Option e - 0.5597
Step-by-step explanation:
Given : Suppose that a random sample of size 49 is to be selected from a population with mean 49 and standard deviation 6.
To find : What is the approximate probability that will be more than 0.5 away from the population mean?
Solution :
Applying z-score formula,
[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
Here, [tex]\mu=49,\sigma=6, n=49[/tex]
As, probability that will be more than 0.5 away from the population mean
i.e. x=49.5,
[tex]z=\frac{49.5-49}{\frac{6}{\sqrt{49}}}[/tex]
[tex]z=\frac{0.5}{0.857}[/tex]
[tex]z=0.58[/tex]
In z-score table the value of z at 0.58 is 0.7190.
x=48.5,
[tex]z=\frac{48.5-49}{\frac{6}{\sqrt{49}}}[/tex]
[tex]z=\frac{-0.5}{0.857}[/tex]
[tex]z=-0.58[/tex]
In z-score table the value of z at -0.58 is 0.2810.
Now, probability that will be more than 0.5 away from the population mean is given by,
[tex]P=P(X>49.5)+P(X<48.5)[/tex]
[tex]P=1-P(X<49.5)+P(X<48.5)[/tex]
[tex]P=1-0.7190+0.2810[/tex]
[tex]P=0.562[/tex]
Therefore, option e is correct.
A hot-air balloon is 140 ft140 ft above the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 60 mi divided by hr60 mi/hr (88 ft divided by s88 ft/s). If the balloon rises vertically at a rate of 14 ft divided by s14 ft/s, what is the rate of change of the distance between the motorcycle and the balloon 10 seconds10 seconds later?
Answer:
Step-by-step explanation:
Given
Balloon velocity is 14 ft/s upwards
Distance between balloon and cyclist is 140 ft
Velocity of motor cycle is 88 ft/s
After 10 sec
motorcyclist traveled a distance of [tex]d_c=88\times 10=880 ft[/tex]
Distance traveled by balloon in 10 s
[tex]d_b=14\times 10=140 ft[/tex]
net height of balloon from ground =140+140=280 ft[/tex]
at [tex]t=10 s[/tex]
distance between cyclist and balloon is [tex]z=\sqrt{280^2+880^2}[/tex]
[tex]z=923.47 ft[/tex]
now suppose at any time t motorcyclist cover a distance of x m and balloon is at a height of h m
thus [tex]z^2=(88t)^2+(14t+140)^2[/tex]
differentiating w.r.t time
[tex]\Rightarrow z\frac{\mathrm{d} z}{\mathrm{d} t}=14\cdot \left ( 14t+140\right )+88\cdot \left ( 88t\right )[/tex]
[tex]\Rightarrow at\ t=10 s[/tex]
[tex]\Rightarrow \frac{\mathrm{d} z}{\mathrm{d} t}=\frac{14\cdot \left ( 14t+140\right )+88\cdot \left ( 88t\right )}{z}[/tex]
[tex]\Rightarrow \frac{\mathrm{d} z}{\mathrm{d} t}=\frac{14\times 280+88^2\times 10}{923.471}[/tex]
[tex]\Rightarrow \frac{\mathrm{d} z}{\mathrm{d} t}=\frac{81,360}{923.471}=88.102\ ft/s[/tex]
Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. A survey of 865 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct the 95% confidence interval for the true proportion of all voters in the state who favor approval. A. 0.444 < p < 0.500 B. 0.438 < p < 0.505 C. 0.471 < p < 0.472 D. 0.435 < p < 0.508
We can be 95% confident that the true proportion of all voters in the state who favor approval lies between 0.438 < p < 0.505. Option B is the correct answer.
Here's a breakdown of the steps involved in constructing the confidence interval:
Calculate the sample proportion (P):
P = 408 (voters favoring approval) / 865 (total voters surveyed) = 0.4714
Determine the critical value (z):
For a 95% confidence interval, the critical value from the standard normal distribution is 1.96.
Calculate the margin of error:
Margin of error = z * √(P * (1 - P) / n)
Margin of error = 1.96 * √(0.4714 * (1 - 0.4714) / 865) = 0.0335
Construct the confidence interval:
Lower bound = P - margin of error = 0.4714 - 0.0335 = 0.4379
Upper bound = P + margin of error = 0.4714 + 0.0335 = 0.5049
Round the bounds to three decimal places:
0.438 < p < 0.505
Therefore, we can be 95% confident that the true proportion of all voters in the state who favor approval lies between 0.438 and 0.505.
A large room has a temperature of 70 degrees Fahrenheit. When a cup of tea is brought into the room, the tea has a temperature of 120 degrees Fahrenheit. The function f(t)=Ce(−kt)+70 represents the situation, where t is time in minutes, C is a constant, and k is a constant. After 3 minutes the cup of tea has a temperature of 100 degrees. What will be the temperature of the tea, in degrees Fahrenheit, after 5 minutes? Round your answer to the nearest tenth, and do not include units.
Answer:
91.3
Step-by-step explanation:
To find the temperature of the tea after 5 minutes, use the given function and the information given in the question. By solving the function for the constant C, you can then find the temperature of the tea after 5 minutes.
Explanation:To find the temperature of the tea after 5 minutes, we need to use the given function and the information given in the question.
First, we can use the given function f(t)=Ce(−kt)+70 to find the value of C. Since we know that after 3 minutes the temperature of the tea is 100 degrees, we can substitute t=3 and T=100 into the function to get:
100 = Ce(−3k)+70
Next, we can use the same function to find the temperature of the tea after 5 minutes. Substituting t=5 into the function and using the value of C that we found earlier, we get:
T = Ce(−5k)+70
Rounding to the nearest tenth, the temperature of the tea after 5 minutes will be approximately 89.4 degrees Fahrenheit.
Ok this time I will fix my English so here I go...
Jonas practices guitar 2 hours a day for 2 years. Bradley practices the guitar 5 hours a day more then Jonas. How many more minutes does Bradley practices the guitar the Jonas over the course of 3 years?
Answer:Bradley practices for 328500 minutes more than Jonas
Step-by-step explanation:
Jonas practices guitar 2 hours a day for 2 years. 24 hours make a day. Assuming there are 365 days in a year.
If he plays 2 hours for 2 days
Number of hours for which he practiced for 365 days would be (365×2)/2 = 365 hours.
Over the course of 3 years, number of hours for which he practiced would be 365×3 = 1095 hours
Bradley practices the guitar 5 hours a day more then Jonas. Jonas practiced for 1 hour a day. This means that Bradley practices the guitar for 5+1 = 6 hours a day. In a year, it would be 6×365 = 2190 hours. In 3 years, it would be 3×2190 = 6570 hours
Difference in number of hours between Bradley and Jonas is
6570 - 1095 = 5475 hours. Since 60 minutes = 1 hour,
5475 hours would be
5475×60 = 328500 minutes.
Researchers are studying the relationship between honesty, age, and self-control conducted an experiment on 160 children between the ages of 5 and 15. The researchers asked each child to toss a fair coin in private and to record the outcome (heads or tails) on a paper sheet, and said they would only reward children who report heads. Half the students were explicitly told not to cheat and the others were not given any explicit instructions. Differences were observed in the cheating rates in the instruction and no instruction groups, as well as some differences across children's characteristics within each group.
a) Identify the population of interest in the study.
A. 80 children between the ages of 5 and 15 who were told not to cheat
B. 160 children between the ages of 5 and 15
C. The researchers
D. All children between the ages of 5 and 15
b) Identify the sample for this study.
A. 80 children between the ages of 5 and 15 who were told not to cheat
B. The researchers
C. All children between the ages of 5 and 15
D. 160 children between the ages of 5 and 15
c) Can the results of the study can be generalized to the population? Should the findings of the study can be used to establish causal relationships.
A. If the sample is randomly selected and representative of the entire population, then the results can be generalized to the target population. Furthermore, since this study is observational, the findings can be used to infer causal relationships.
B. If the sample is randomly selected and representative of the entire population, then the results can be generalized to the target population. Furthermore, since this study is experimental, the findings can be used to infer causal relationships.
C. If the sample is randomly selected and representative of the entire population, then the results can be generalized to the target population. Furthermore, since this study is experimental, the findings cannot be used to infer causal relationships.D. If the sample is randomly selected and representative of the entire population, then the results can be generalized to the target population. Furthermore, since this study is observational, the results cannot be used to infer causal relationships.
Answer:
Step-by-step explanation:
Given that researchers are studying the relationship between honesty, age, and self-control conducted an experiment on 160 children between the ages of 5 and 15.
a) The population is the children between the ages of 5 and 15 in general
D. All children between the ages of 5 and 15
b) Sample is
D. 160 children between the ages of 5 and 15
c) Yes becauB. If the sample is randomly selected and representative of the entire population, then the results can be generalized to the target population. Furthermore, since this study is experimental, the findings can be used to infer causal relationships.se randomly selected and also sample size is above 30
The population of interest is all children aged 5 to 15; the study's sample consists of the 160 participating children. Results can be generalized if the sample is representative, and as an experimental study, it can establish causal relationships.
Explanation:The population of interest in the study is D. All children between the ages of 5 and 15 because the researchers are attempting to understand a broader phenomenon that could be applicable to all children within this age range, not just those in the study. The sample for the study is D. 160 children between the ages of 5 and 15 because these are the actual participants that researchers have collected data from within this experiment.
Regarding the generalizability and establishment of causal relationships of the study, the correct option is B. If the sample is randomly selected and representative of the entire population, then the results can be generalized to the target population. Furthermore, since this study reports the manipulation of an independent variable (the instruction given to one group and not to the other), it qualifies as an experimental study, and its results can inform us about causal relationships.