Answers
Step-by-step explanation:
∑ (4ⁿ⁺¹ / 5ⁿ)
Rewrite 4ⁿ⁺¹ as 4 (4ⁿ).
∑ 4 (4ⁿ / 5ⁿ)
∑ 4 (4/5)ⁿ
This is a geometric series. The sum of an infinite geometric series is:
S = a / (1 −r)
where a is the first term and r is the common ratio.
Here, the first term is 16/5 (because n starts at 1), and the common ratio is 4/5.
S = 16/5 / (1−4/5)
S = 16/5 / (1/5)
S = 16
Related Questions
The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 80% confident that his estimate is within six percentage points of the true populationpercentage?
Complete parts (a) through (c) below.
1) Assume that nothing is known about the percentage of adults who have heard of the brand.
a.n=_________(Round up to the nearest integer.)
2) Assume that a recent survey suggests that about 85% of adults have heard of the brand.
b.n=_________(Round up to the nearest integer.)
3) Given that the required sample size is relatively small, could he simply survey the adults at the nearest college?
Answers
Answer:
1) n=114
2) n=59
3) On this case no, because if we survey just the adults of the nearest college that would be a convenience sample. And when we use "convenience sample" we have some problems associated to bias. This methodology it's not appropiate in order to have a good estimation of the parameter of interest. It's better use a random, cluster or stratified sampling.
Step-by-step explanation:
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".
The population proportion have the following distribution
[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]
In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 80% of confidence, our significance level would be given by [tex]\alpha=1-0.80=0.2[/tex] and [tex]\alpha/2 =0.1[/tex]. And the critical value would be given by:
[tex]z_{\alpha/2}=-1.28, z_{1-\alpha/2}=1.28[/tex]
Part 1
The margin of error for the proportion interval is given by this formula:
[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex] (a)
And on this case we have that [tex]ME =\pm 0.06[/tex] or 6% points, and we are interested in order to find the value of n, if we solve n from equation (a) we got:
[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex] (b)
Since we don't have a prior estimate of [tex]\het p[/tex] we can use 0.5 as a good estimate, replacing into equation (b) the values from part a we got:
[tex]n=\frac{0.5(1-0.5)}{(\frac{0.06}{1.28})^2}=113.77[/tex]
And rounded up we have that n=114
Part 2
On this case we have a prior estimate for the population proportion and is [tex]\hat p =0.85[/tex] so replacing the values into equation (b) we got:
[tex]n=\frac{0.85(1-0.85)}{(\frac{0.06}{1.28})^2}=58.027[/tex]
And rounded up we have that n=59
Part 3
On this case no, because if we survey just the adults of the nearest college that would be a convenience sample. And when we use "convenience sample" we have some problems associated to bias. This methodology it's not appropiate in order to have a good estimation of the parameter of interest. It's better use a random, cluster or stratified sampling.
The number of customers that enter a store during one day in an example of :
-a continuous random variable
-a discrete random variable
-either a continuous or a discrete random variable, depending on the number of the customers
-either a continuous or a discrete random variable, depending on the gender of the customers
Answers
Answer:
a discrete random variable
Step-by-step explanation:
You can only have a natural number of clients entering the store.
For example, 0 clients, 1 client, 2 clients, 100 clients, ...
You cannot have a decimal value, for example, 0.5 clients.
So the correct answer is:
a discrete random variable
A researcher wishes to determine the average number of vehicles are registered to a typical Houston residence. In order to do this, he sends a survey to 250 randomly selected residences asking for them to indicate the number of registered and return the survey. Identify the population.
Answers
Answer: Houston residences
Step-by-step explanation:
A population is the group of members comes under the same criteria by the researcher's point of view.Here , The objective of the researcher is to determine the average number of vehicles are registered to a typical Houston residence.
Clearly , the population is this situation is "Houston residences" having vehicles.
Note : 250 randomly selected residences are defining the sample of the entire population of Houston residences which is a subset of population.
Hence, the correct answer is Houston residences.
Say you're playing three-card poker; that is, you're dealt three cards in a row at random from a standard deck of 52 cards. What are the odds of getting a pair or three of a kind?
Answers
Answer:
Step-by-step explanation:
Given
There are 52 cards in total
there are total of 13 pairs of same cards with each pair containing 4 cards
Probability of getting a pair or three of kind card=1-Probability of all three cards being different
Probability of selecting all three different cards can be find out by selecting a card from first 13 pairs and remaining 2 cards from remaining 12 pairs i.e.
[tex]=\frac{52\times 48\times 44}{52\times 51\times 50}[/tex]
for first card there are 52 options after choosing first card one pair is destroyed as we have to select different card .
For second card we have to select from remaining 12 pairs i.e. 48 cards and so on for third card.
Required Probability is [tex]=1-\frac{52\times 48\times 44}{52\times 51\times 50}[/tex]
[tex]=\frac{22776}{132600}[/tex]
The mean consumption of bottled water by a person in the United States is 28.5 gallons per year. You believe that a person consume more than 28.5 gallons in bottled water per year. A random sample of 100 people in the United States has a mean bottled water consumption of 27.8 gallons per year with a standard deviation of 4.1 gallons. At α = 0.10 significance level can you reject the claim?
Answers
Answer:
[tex]t=\frac{27.8-28.5}{\frac{4.1}{\sqrt{100}}}=-1.707[/tex]
[tex]p_v =P(t_{99}<-1.707)=0.0455[/tex]
If we compare the p value with a significance level for example [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we reject the null hypothesis, so there is not enough evidence to conclude that the mean for the consumption is less than 28.5 gallons at 0.1 of significance, so we can reject the claim that person consume more than 28.5 gallons.
Step-by-step explanation:
Data given and notation
[tex]\bar X=27.8[/tex] represent the mean for the account balances of a credit company
[tex]s=4.1[/tex] represent the population standard deviation for the sample
[tex]n=1000[/tex] sample size
[tex]\mu_o =28.5[/tex] represent the value that we want to test
[tex]\alpha=0.1[/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)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to determine if the mean for the person consume is more than 28.5 gallons, the system of hypothesis would be:
Null hypothesis:[tex]\mu \geq 28.5[/tex]
Alternative hypothesis:[tex]\mu < 28.5[/tex]
We don't know the population deviation, so for this case we can use the 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".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]t=\frac{27.8-28.5}{\frac{4.1}{\sqrt{100}}}=-1.707[/tex]
Calculate the P-value
First we need to calculate the degrees of freedom given by:
[tex]df=n-1=100-1=99[/tex]
Since is a one-side lower test the p value would be:
[tex]p_v =P(t_{99}<-1.707)=0.0455[/tex]
In Excel we can use the following formula to find the p value "=T.DIST(-1.707,99)"
Conclusion
If we compare the p value with a significance level for example [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we reject the null hypothesis, so there is not enough evidence to conclude that the mean for the consumption is less than 28.5 gallons at 0.1 of significance, so we can reject the claim that person consume more than 28.5 gallons.
Which of the following statements are true of hypothesis tests?
1.You must state null and alternative hypotheses in the context of the problem.
2.You must state a significance level so you can decide if a given P-value gives you evidence to reject the null hypothesis.
3.You must state a conclusion in the context of the problem.
Answers
In hypothesis testing, it is critical to state the null and alternative hypotheses, choose an appropriate significance level, and conclude in the context of the problem. Decisions must reflect the probabilistic nature of the tests, with careful consideration of Type I and Type II errors.
In hypothesis testing, the following statements are indeed true:
You must state null and alternative hypotheses in the context of the problem.
You must state a significance level so you can decide if a given P-value provides evidence to reject the null hypothesis.
You must state a conclusion in the context of the problem.
When conducting a hypothesis test, one must also be mindful not to claim that a hypothesis is definitively proven true or false due to the probabilistic nature of hypothesis testing. Instead, you can infer whether there is sufficient evidence to support the alternative hypothesis if the null hypothesis is rejected. However, remember that making a decision at a certain significance level involves a trade-off between Type I and Type II errors.
A fisherman catches fish according to a Poisson process with rate lambda = 0.6 per hour. The fisherman will keep fishing for two hours. At the end of the second hour, if he has caught at least one fish, he quits; Otherwise, he continues until he catches one fish. (a) Find the probability that he stays for more than two hours. (b) Find the probability that the total time he spends fishing is between two and five hours. (c) Find the expected number offish that he catches. (d) Find the expected total fishing time, given that he has been fishing for four hours.
Answers
The probability that the fisherman stays for more than two hours is approximately 0.4512. The probability that the total time the fisherman spends fishing is between two and five hours is approximately 0.5043. The expected number of fish that the fisherman catches is 0.6.
Explanation:To solve this problem, we can use the concept of a Poisson process and the properties of the Poisson distribution.
(a) We want to find the probability that the fisherman stays for more than two hours. Since the fisherman quits at the end of the second hour if he has caught at least one fish, he will stay for more than two hours only if he hasn't caught any fish in the first two hours. Using the Poisson distribution, the probability of catching zero fish in two hours is given by P(X=0)=e^(-lambda*t)*(lambda^0)/(0!)=e^(-0.6*2)≈0.5488. Therefore, the probability that the fisherman stays for more than two hours is 1 - P(X=0) = 1 - 0.5488 ≈ 0.4512.
(b) The total time the fisherman spends fishing can be between two and five hours. This can happen in three ways: fishing for 2 hours without catching fish (P(X=0)) and then fishing for 3 more hours until catching the first fish (P(X=1)); fishing for 3 hours without catching fish (P(X=0)) and then fishing for 2 more hours until catching the first fish (P(X=1)); fishing for 4 hours without catching fish (P(X=0)) and then fishing for 1 more hour until catching the first fish (P(X=1)). Using the Poisson distribution, we can calculate the probabilities for each case: P(X=0) = e^(-0.6*2) ≈ 0.5488, P(X=1) = e^(-0.6*3)*(0.6^1)/(1!) ≈ 0.3293. Adding up these probabilities, we get P(X=0)*P(X=1) + P(X=0)*P(X=1) + P(X=0)*P(X=1) = 0.5488*0.3293 + 0.5488*0.3293 + 0.5488*0.3293 ≈ 0.5043. Therefore, the probability that the total time the fisherman spends fishing is between two and five hours is approximately 0.5043.
(c) The expected number of fish that the fisherman catches can be found using the formula for the mean of a Poisson distribution, which is equal to the rate parameter lambda. In this case, lambda = 0.6, so the expected number of fish is 0.6.
(d) To find the expected total fishing time given that the fisherman has been fishing for four hours, we need to condition on the event that the fisherman hasn't caught any fish in the first four hours. Using the Poisson distribution, the probability of catching zero fish in four hours is given by P(X=0) = e^(-lambda*t)*(lambda^0)/(0!) = e^(-0.6*4) ≈ 0.3012. Therefore, the expected total fishing time given that the fisherman has been fishing for four hours is 4 + (1/P(X=0)) = 4 + (1/0.3012) ≈ 7.32 hours.
Learn more about Probability and Statistics here:https://brainly.com/question/27342429
#SPJ11
......Help Please......
Answers
Answer:
Step-by-step explanation:
the sign here means division
7 divided by the number in the box equals to 8
simply meaning 7 multiplied by 8 will give the number
7 x 8 = 56
In many population growth problems, there is an upper limit beyond which the population cannot grow. Many scientists agree that the earth will not support a population of more than 16 billion. There were 2 billion people on earth in 1925 and 4 billion in 1975. If is the population years after 1925, an appropriate model is the differential equationdy/dt=ky(16-y)Note that the growth rate approaches zero as the population approaches its maximum size. When the population is zero then we have the ordinary exponential growth described by y'=16ky. As the population grows it transits from exponential growth to stability.(a) Solve this differential equation.(b) The population in 2015 will be(c) The population will be 9 billion some time in the year
Answers
Answer:
a) (y-16)/y = -7*e∧(-0.016946*t)
b) y = 6.34
c) t = 129.66 years in 2055
Step-by-step explanation:
a) dy/dt = ky*(16-y)
Solving the differential equation we have
dy / (y*(y-16)) = -k dt
∫ dy / (y*(y-16)) = ∫ -k dt
(-1/16)*Ln (y) + (1/16)*Ln (y-16) = -k*t + C
(1/16) Ln ((y-16)/y) = -k*t + C
Ln ((y-16)/y) = -16*k*t + C
(y-16)/y = C*e∧(-16*k*t)
If t = 0 and y = 2
(2-16)/2 = C*e∧(0)
C = -7 then we have
(y-16)/y = -7*e∧(-16*k*t)
In 1975 we have t = 1975 - 1925= 50 years and y = 4
(4-16)/4 = -7*e∧(-16*k*50)
k= - Ln (3/7) / 800 = 0.001059
Finally, the differential equation will be
(y-16)/y = -7*e∧(-16*0.001059*t)
(y-16)/y = -7*e∧(-0.016946*t)
b) In 2015 we have t = 2015 – 1925 = 90 years
(y-16)/y = -7*e∧(-0.016946*90)
Solving the equation we get
y = 6.34
c) If y = 9
(9-16)/9 = -7*e∧(-0.016946*t)
t = 129.66 years in 2055
The solution for (a)[tex]a) (y-16)/y = -7*e^{(-0.016946*t)}[/tex]b) y = 6.34 and (c) the value of t is 129.66 years in 2055
We have given that,
[tex]a) dy/dt = ky*(16-y)[/tex]
By using variable separable form we have,
What is the variable separable form?A variable separable differential equation is any differential equation in which variables can be separated
Therefore by solving the differential equation we have
[tex]dy / (y*(y-16)) = -k dt[/tex]
integrating both side with respect to t
[tex]\int dy / (y*(y-16)) = \int -k dt[/tex]
Solve the integration of the above
[tex](-1/16)*ln (y) + (1/16)*ln (y-16) = -k*t + C[/tex]
[tex](1/16) ln ((y-16)/y) = -k*t + C[/tex]
[tex]ln ((y-16)/y) = -16*k*t + C[/tex]
[tex](y-16)/y = C*e^{(-16*k*t)}[/tex]
If t = 0 and y = 2
[tex](2-16)/2 = C*e^{0}[/tex]
C = -7 then we have
[tex](y-16)/y = -7*e^{(-16*k*t)}[/tex]
In 1975 we have t = 1975 - 1925= 50 years and y = 4
[tex](4-16)/4 = -7*e^{(-16*k*50)[/tex]
[tex]k= - Ln (3/7) / 800 = 0.001059[/tex]
Finally, the differential equation will be
[tex](y-16)/y = -7*e^{(-16*0.001059*t)}[/tex]
[tex](y-16)/y = -7*e^{(-0.016946*t)}[/tex]
b) In 2015 we have t = 2015 – 1925 = 90 years
[tex](y-16)/y = -7*e^{(-0.016946*90)}[/tex]
Solving the equation we get
y = 6.34
c) If y = 9
[tex](9-16)/9 = -7*e^{(-0.016946*t)}[/tex]
Therefore we get the value of t is 129.66 years in 2055
To learn more about the differential equation visit:
https://brainly.com/question/1164377
The following questions refer to the CRT Technologies project selection example presented in this chapter. Formulate a constraint to implement the conditions described in each of the following statements. a. Out of projects 1, 2, 4, and 6, CRT’s management wants to select exactly two projects. b. Project 2 can be selected only if project 3 is selected and vice-versa. c. Project 5 cannot be undertaken unless both projects 3 and 4 are also undertaken. d. If projects 2 and 4 are undertaken, then project 5 must also be undertaken
Answers
This ensures that if both project 2 and project 4 are selected, project 5 must also be selected. Since x5 is a binary variable, 2x5 is equivalent to x5, ensuring that project 5 is selected when the condition is met. (option d)
Let's formulate constraints for each of the given statements:
a. Out of projects 1, 2, 4, and 6, CRT’s management wants to select exactly two projects.
Let [tex]\(x_i\)[/tex] be a binary decision variable representing whether project i is selected or not, where [tex]\(i = 1, 2, 4, 6\).[/tex]
The constraint can be formulated as:
[tex]\[x_1 + x_2 + x_4 + x_6 = 2\][/tex]
This ensures that exactly two out of the listed projects are selected.
b. Project 2 can be selected only if project 3 is selected and vice-versa.
Let [tex]\(x_2\) and \(x_3\)[/tex] be binary decision variables representing whether project 2 and project 3 are selected, respectively.
The constraints can be formulated as:
[tex]\[x_2 \leq x_3\]\[x_3 \leq x_2\][/tex]
These constraints ensure that if project 2 is selected, project 3 must also be selected, and vice versa.
c. Project 5 cannot be undertaken unless both projects 3 and 4 are also undertaken.
Let [tex]\(x_3\), \(x_4\), and \(x_5\)[/tex] be binary decision variables representing whether project 3, project 4, and project 5 are selected, respectively.
The constraint can be formulated as:
[tex]\[x_5 \leq x_3 + x_4\][/tex]
This ensures that if project 5 is selected, both project 3 and project 4 must also be selected.
d. If projects 2 and 4 are undertaken, then project 5 must also be undertaken.
Let [tex]\(x_2\), \(x_4\), and \(x_5\)[/tex] be binary decision variables representing whether project 2, project 4, and project 5 are selected, respectively.
The constraint can be formulated as:
[tex]\[x_2 + x_4 \leq 2x_5\][/tex]
Suppose the lengths, in seconds, of the songs in an online database are normally distributed. For a random sample of songs, the confidence interval (184.00, 188.00) is generated. Find the sample mean x. Give just a number for your answer. For example, if you found that the sample mean was 12, you would enter 12.
Answers
Answer:
186
Step-by-step explanation:
The sample mean, x, is the central value in the confidence interval, that is, the average between the upper and lower bounds of the interval.
In this case, the Lower bound is 184.00 and the upper bound is 188.00. Therefore, the sample mean is given by:
[tex]x = \frac{184+188}{2}\\x=186.00[/tex]
The mean length, in seconds, of the sampled songs is 186.00.
The sample mean is 186.
The sample mean, denoted as [tex]\( x \)[/tex], can be found by taking the midpoint of the confidence interval. In this case, the confidence interval is (184.00, 188.00). To find [tex]\( x \)[/tex], we take the average of the lower and upper bounds of the interval. Thus,
[tex]\[ x = \frac{184.00 + 188.00}{2} = \frac{372.00}{2} = 186.00 \][/tex]
Therefore, the sample mean [tex]\( x \)[/tex] is 186.00 seconds. This means that, on average, the length of songs in the online database, based on the given sample, is 186.00 seconds.
A numerical description of the outcome of an experiment is called a
a. descriptive statistic.
b. probability function.
c. variance.
d. random variable.
Answers
Answer: d. random variable.
Step-by-step explanation:
A random variable is a numerical description of outcomes of an experiment, it can be used to represent the possible values of a past experiment or yet-to-be-performed experiments. It is a variable whose values depends on outcome of a random occurrence. Random variables also allows the calculation of probability of an occurrence or result in a particular experiment.
The numerical description of the outcome of an experiment is best described as a random variable. It is not referred to as a descriptive statistic, a probability function, or variance.
Explanation:In statistics, a numerical description of the outcome of an experiment is referred to as a random variable. The term random variable refers to a function that assigns a real number to each outcome of an experiment conducted according to a certain probability distribution. This term is central to probability theory and statistics, in which numerical results of random variables are analyzed to understand underlying processes or to make predictions.
On the other hand, descriptive statistics summarize and organize characteristics of a data set. A probability function is a mathematical function that provides the probabilities of occurrence of different possible outcomes. Variance is a measurement of spread between numbers in a data set.
Learn more about random variable here:https://brainly.com/question/33194053
#SPJ6
Compute Δy and dy for the given values of x and dx = Δx. (Round your answers to three decimal places.) y = x , x = 1, Δx = 1 Δy = dy =
Answers
Answer:
Δy = 1
dy = 1
Step-by-step explanation:
Data provided in the question:
dx = Δx
y = x
x = 1,
Δx = 1
Now,
we know,
Δy = f( x + Δx ) - f(x)
also, we have
y = f(x) = x
thus,
f( x + Δx ) = x + Δx
Therefore,
Δy = ( x + Δx ) - x
on substituting the respective values, we get
Δy = ( 1 + 1 ) - 1
or
Δy = 1
and,
dy = f'(x) = [tex]\frac{d(x)}{dx}[/tex]
or
dy = 1
Nationwide, the average waiting time until a electric utility customer service representative answers a call is 200 seconds per call. The Gigantic Kilowatt Energy Company took a sample of 30 calls and found that, on the average, they answered in 120 seconds per call. Moreover, it is know that the standard deviation of the times for all such calls is 25 seconds. At the .05 significance level, is there evidence that this company's mean response time is lower than the average utility?
Answers
Answer:
[tex]z=\frac{120-200}{\frac{25}{\sqrt{30}}}=-17.527[/tex]
[tex]p_v =P(Z<-17.527) \approx 0[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis.
We can say that at 5% of significance the mean average waiting time is significantly less than 200 seconds per call.
Step-by-step explanation:
Data given and notation
[tex]\bar X=120[/tex] represent the sample mean
[tex]\sigma=25[/tex] represent the population standard deviation
[tex]n=30[/tex] sample size
[tex]\mu_o =200[/tex] represent the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the population mean is less than 200, the system of hypothesis are :
Null hypothesis:[tex]\mu \geq 200[/tex]
Alternative hypothesis:[tex]\mu < 200[/tex]
Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
z-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".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]z=\frac{120-200}{\frac{25}{\sqrt{30}}}=-17.527[/tex]
P-value
Since is a one-side left tailed test the p value would given by:
[tex]p_v =P(Z<-17.527) \approx 0[/tex]
Conclusion
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis.
We can say that at 5% of significance the mean average waiting time is significantly less than 200 seconds per call.
Zener cards are often used to test the "psychic ability" of individuals. In the Zener deck, there are five different patterns displayed, and each has a 1/5 probability of being drawn from a well-shuffled deck. The five patterns are: circle, plus sign, wavy lines, empty box, and star. One hundred trials were conducted and your very impressive friend guessed right on 41 of those trials. Given this sample, can we use the normal approximation to the binomial?
Answers
Answer:
We need to check the conditions in order to use the normal approximation.
[tex]np=100*0.2=20 \geq 10[/tex]
[tex]n(1-p)=100*(1-0.2)=80 \geq 10[/tex]
If we check the conditions with the estimated proportion we got:
[tex]n\hat p=100*0.41=41 \geq 10[/tex]
[tex]n(1-\hat p)=100*(1-0.41)=59 \geq 10[/tex]
So we see that we satisfy the conditions and then we can apply the approximation.
Step-by-step explanation:
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=100, p=0.2)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
We need to check the conditions in order to use the normal approximation.
[tex]np=100*0.2=20 \geq 10[/tex]
[tex]n(1-p)=100*(1-0.2)=80 \geq 10[/tex]
If we check the conditions with the estimated proportion we got:
[tex]n\hat p=100*0.41=41 \geq 10[/tex]
[tex]n(1-\hat p)=100*(1-0.41)=59 \geq 10[/tex]
So we see that we satisfy the conditions and then we can apply the approximation.
a tree casts a shadow 8 feet long. A 6-foot Man cast a shadow 4 feet long. The triangle formed by the tree and its shadow is similar to the triangle formed by the man and his shadow. How tall is the tree?
Answers
Answer:
12 feet
Step-by-step explanation:
Draw a diagram (see picture below). The tree and its shadow is one triangle, the man and its shadow is another triangle. We assume both are right triangles because people and trees stand vertical.
Create a proportion to solve. Put the missing value in a numerator.
Tree height / Tree shadow = Man height / Man shadow
[tex]\frac{x}{8} =\frac{6}{4}[/tex]
Solve using cross multiplication. Multiply x by 4. Multiply 6 by 8.
4x = 48 Divide both sides by 4 to isolate x.
x = 12 Height of tree
The tree is 12 feet tall.
The probability density function f(x) of a random variable X that has a uniform distribution between a and b is:
-(b + a)/2
-(a − b)/2
-1/b − 1/a
-None of these choices.
Answers
The probability density function (pdf) is;
f(x)=[tex]\frac{1}{b - a}[/tex] for a ≤ x ≤ b.
while the mean is given as [tex]\frac{a + b}{2}[/tex]
And the standard deviation given as [tex]\sqrt{\frac{(b - a)^{2} }{12} }[/tex]
Answer: -None of these choices.
PDF = 1/(b-a)
Step-by-step explanation:
The probability density function f(x) of a random variable X that has a uniform distribution between a and b is given by:
PDF = 1/(b-a) for X€[a,b]
otherwise zero.
Robin Inc. feared that the average company loss is running beyond $34,000. It initially conducted a hypothesis test on a sample extracted from its database. The hypothesis was formulated as H0: average company loss ≤ $34,000 H1: average company loss > $34,000. The test resulted in favor of Robin Inc.'s loss not exceeding $34,000. Detailed study of company accounts later revealed that the average company loss had run up to $37,896. Which of the following errors were made during the hypothesis test? Select one: a) Type III error b) Type II error c) Type I error d) Type IV error
Answers
Answer
b. Type II error
Step-by-step explanation:
The hypothesis was formulated as
the solution can be seen in the attached document
There is a strong correlation between the temperature and the number of skinned knees on playgrounds. Does this tell us that warm weather causes children to trip? Choose the correct answer below. A. Yes. In warm weather, more children will go outside and play. B. No. Warm weather will cause less children to trip and suffer skinned knees. C. No. In warm weather, more children will go outside and play. D. Yes. Warm weather will cause more children to trip and suffer skinned knees.
Answers
Answer:
C. No. In warm weather, more children will go outside and play
Step-by-step explanation:
The correct answer is option C: No. In warm weather, more children will go outside and play.
Correlation means that two variables are related, but it does not necessarily mean that one causes the other. In this case, there is a strong correlation between temperature and the number of skinned knees on playgrounds, but it does not tell us that warm weather causes children to trip.
Option A is incorrect because warm weather does not directly cause children to go outside and play. It may be a factor, but it is not the sole reason. Option B is incorrect because warm weather does not cause fewer children to trip and suffer skinned knees. In fact, the correlation suggests that more children are likely to be outside playing in warm weather, which could potentially increase the number of skinned knees. Option D is incorrect because warm weather does not directly cause more children to trip and suffer skinned knees.
The correlation suggests that the increase in skinned knees is likely due to more children being outside and playing, rather than the warm weather itself. To summarize, the strong correlation between temperature and the number of skinned knees on playgrounds indicates that in warm weather, more children will go outside and play. However, it does not tell us that warm weather causes children to trip.
To know more about temperature here
https://brainly.com/question/27944554
#SPJ2
Solve the following equation. log Subscript 2 Baseline (3 x plus 7 )equals 5 The solution set is StartSet nothing EndSet . (Simplify your answer.)
Answers
Answer: [tex]x=\dfrac{25}{3}[/tex]
Step-by-step explanation:
The given equation : [tex]\log_2(3x+7)=5[/tex]
Using logarithmic property : [tex]\log_a N=M\to N=a^M[/tex]
The given equation will be equivalent to [tex](3x+7)=2^5[/tex]
[tex]\Rightarrow\ 3x+7=32[/tex]
Subtract 7 from both sides , we get
[tex]3x=25[/tex]
Divide both sides by 3 , we get
[tex]x=\dfrac{25}{3}[/tex]
Hence, the solution is [tex]x=\dfrac{25}{3}[/tex]
PLEASE HELP ME!!!
What transformations are represented by the following coordinate graphing? (geometry)
(a,b) --> (a,-b)
(a,b) --> (a, b+5)
(a,b) --> (b,-a)
Answers
Step-by-step explanation:
(a, b) → (a, -b)
This is a reflection across the x-axis.
(a, b) → (a, b+5)
This is a translation 5 units up.
(a, b) → (b, -a)
This is a rotation of 270° about the origin.
Clare is using little wooden cubes with edge length 1/2 inch to build a larger cube that has edge length 4 inches. How many little cubes does she need? explain your reasoning
Answers
To form a larger 4-inch cube, Clare will need 512 wooden cubes with an edge length of 1/2 inch. We find this by dividing the volume of the large cube (64 cubic inches) by the volume of the small one (1/8 cubic inch).
Explanation:Clare is building a larger cube with an edge length of 4 inches, consisting of smaller wooden cubes each with an edge length of 1/2 inch. To solve this problem, we need to find out how many smaller cubes make up the volume of the larger cube. The volume of a cube is found by multiplying the length of an edge by itself three times, or cube the edge length.
So first, we calculate the volume of the large cube which is 4in * 4in * 4in = 64 cubic inches.
Next, we calculate the volume of a small cube which is (1/2in) * (1/2in) * (1/2in) = 1/8 cubic inch.
Finally, we divide the volume of the large cube by the volume of the small cube to find out how many small cubes are needed. Thus, 64 cubic inches / 1/8 cubic inch = 512. So, Clare will need 512 small wooden cubes to construct her larger cube.
Learn more about Volume of Cubes here:https://brainly.com/question/23526372
#SPJ12
To find the number of little cubes needed, divide the volume of the larger cube by the volume of each little cube.
Explanation:To find the number of little cubes Clare needs to build a larger cube with an edge length of 4 inches, we need to
determine the volume of the larger cube and divide it by the volume of each little cube.
The volume of the larger cube is calculated by multiplying the length of one side by itself three times (4 x 4 x 4 = 64 cubic inches).
The volume of each little cube is calculated by multiplying the length of one side by itself three times (1/2 x 1/2 x 1/2 = 1/8 cubic inches).
To find the number of little cubes needed, we divide the volume of the larger cube by the volume of each little cube (64 / (1/8) = 512 little cubes).
Learn more about Calculating volume of cubes here:https://brainly.com/question/26761255
#SPJ2
Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in [0,2Ï€) and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." cos2(5x)=sin2(5x)
Answers
Answer:
[tex]\large\boxed{x=\pm\dfrac{3\pi}{20}+\dfrac{2n\pi}{5}\ \vee\ x=\pm\dfrac{\pi}{20}+\dfrac{2n\pi}{5}}[/tex]
Step-by-step explanation:
[tex][tex]\cos^2(5x)=\sin^2(5x)\qquad\text{substitute}\ t=5x\\\\\cos^2t=\sin^2t\qquad\text{use}\ \sin^2\theta+\cos^2\theta=1\to\sin^2\theta=1-\cos^2\theta\\\\\cos^2t=1-\cos^2t\qquad\text{add}\ \cos^2t\ \text{to both sides}\\\\2\cos^2t=1\qquad\text{divide both sides by 2}\\\\\cos^2t=\dfrac{1}{2}\Rightarrow \cos t=\pm\sqrt{\dfrac{1}{2}}\\\\\cos t=\pm\dfrac{\sqrt2}{2}\\\\\cos t=-\dfrac{\sqrt2}{2}\Rightarrow t=\pm\dfrac{3\pi}{4}+2n\pi\\\\\cos t=\dfrac{\sqrt2}{2}\Rightarrow t=\pm\dfrac{\pi}{4}+2n\pi[/tex]
[tex]t=5x\\\\5x=\pm\dfrac{3\pi}{4}+2n\pi\ \vee\ 5x=\pm\dfrac{\pi}{4}+2n\pi\qquad\text{divide both sides by 5}\\\\x=\pm\dfrac{3\pi}{20}+\dfrac{2n\pi}{5}\ \vee\ x=\pm\dfrac{\pi}{20}+\dfrac{2n\pi}{5}[/tex]
The trigonometric equation 2cos^2(x) + 2 = 4 has solutions x = 0 and x = π in the interval [0, 2π). To include all possible solutions, we express them as x = 2nπ and x = π + 2nπ with n as any integer.
To solve the trigonometric equation 2cos^2(x) + 2 = 4, we start by simplifying the equation:
Subtract 2 from both sides to get 2cos^2(x) = 2.Divide both sides by 2 to get cos^2(x) = 1.Take the square root of both sides, considering both positive and negative roots, to get cos(x) = ±1.Find angles x in the interval [0, 2π) where the cosine is ±1. For cos(x) = 1, x = 0. For cos(x) = -1, x = π.This equation has two solutions in the interval [0, 2π): x = 0 and x = π (which are 0 and 3.1416 when rounded to four decimal places). To express the infinite set of solutions that occur at every period of cos(x), we add 2nπ to each solution, where n is any integer, giving the final solutions as x = 2nπ and x = π + 2nπ.
the complete Question is given below:
Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in [0, 2π)
and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."
2cos^2 (x) + 2 = 4
Enter your answer in radians, as an exact answer when possible. Multiple answers should be separated by commas.
This question is to show that we can `recode' and model a situation that depends on nitely many past states as a homogeneous Markov chain. Suppose we model the daily weather as a Markov chain. The weather has just two states: cloudy and sunny. Suppose that if it is sunny today and was sunny yesterday then it will be sunny tomorrow with probability 0:6; if sunny today but cloudy yesterday then it will be sunny tomorrow with probability 0:5; if cloudy today but sunny yesterday then it will be sunny tomorrow with probability 0:4; if it was cloudy for the last two days then it will be sunny tomorrow with probability 0:2. Calculate the expected fraction of cloudy days.
Answers
Answer:
F=y+z=4/6.25
Step-by-step explanation:
First, we have to consider that in the problem model we have only two possible states: sunny and cloudy. Now, according to the information given in the statement, we also have the behavior of the last two days. In any case, we can have four possible transitional states:
Today-Yesterday(S=Sunny, C=cloudy)
1) ST and SY (Sunny today and sunny yesterday)
2) ST and CY.
3) CT and SY.
4) CT and CY.
Now, according to the statement, the probabilities given for the four states can be expressed by the following matrix:
[tex]\left[\begin{array}{cccc}0.6&0&(1-0.6)&0\\0.5&0&(1-0.5)&0\\0&0.4&0&(1-0.4)\\0&0.2&0&(1-0.2)\end{array}\right][/tex]
Now, making w, x, y, z as the transition probabilities for the four states mentioned, we then have that:
x=0.6w+0.5x
w=1.25x (1)
x=0.4y+0.2z (2)
y=0.4w+0.5x
y= 0.4(1.25x)+0.5x=x
y=x (3)
replacing 3 in 2:
y=0.4y+0.2x
x=3y (4)
And as w+x+y+z= 1 (no more possible combinations):
w+x+y+z=1 (5)
So, replacing the expressions obtained previously in equation 5, we have finally that:
1.25x+x+x+3x=1
x=1/6.25=y
z=3x=3/6.25
So, the fraction of sunny days is given by:
F=y+z=4/6.25
a 62 year old man owns a non-tax qualified variable annuity. if this indvidual makes a lump-sum withdrawal from the plan, this would:
Answers
Answer:
If the 62 year old man makes a lump-sum withdrawal from the plan or tax structure, his investments would start incurring ordinary income taxes without attracting any other form of penalties. However, it has to be noted that prior to withdrawal of the lump-sum, his investments would grow without incurring income taxes.
Define the double factorial of n, denoted n!!, as follows:n!!={1⋅3⋅5⋅⋅⋅⋅(n−2)⋅n} if n is odd{2⋅4⋅6⋅⋅⋅⋅(n−2)⋅n} if n is evenand (−1)!! =0!! =1.Find the radius of convergence for the given power series.[(8^n*n!*(3n+3)!*(2n)!!)/(2^n*[(n+9)!]^3*(4n+3)!!)]*(8x+6)^n
Answers
Answer:
Radius of convergence of power series is [tex] \lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}=\frac{1}{108}[/tex]
Step-by-step explanation:
Given that:
n!! = 1⋅3⋅5⋅⋅⋅⋅(n−2)⋅n n is odd
n!! = 2⋅4⋅6⋅⋅⋅⋅(n−2)⋅n n is even
(-1)!! = 0!! = 1
We have to find the radius of convergence of power series:
[tex]\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}](8x+6)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}]2^{n}(4x+3)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}](x+\frac{3}{4})^{n}\\[/tex]
Power series centered at x = a is:
[tex]\sum_{n=1}^{\infty}c_{n}(x-a)^{n}[/tex]
[tex]\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}](8x+6)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}]2^{n}(4x+3)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}4^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}](x+\frac{3}{4})^{n}\\[/tex]
[tex]a_{n}=[\frac{8^{n}4^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}]\\\\a_{n+1}=[\frac{8^{n+1}4^{n+1}n!(3(n+1)+3)!(2(n+1))!!}{[(n+1+9)!]^{3}(4(n+1)+3)!!}]\\\\a_{n+1}=[\frac{8^{n+1}4^{n+1}(n+1)!(3n+6)!(2n+2)!!}{[(n+10)!]^{3}(4n+7)!!}][/tex]
Applying the ratio test:
[tex]\frac{a_{n}}{a_{n+1}}=\frac{[\frac{32^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}]}{[\frac{32^{n+1}(n+1)!(3n+6)!(2n+2)!!}{[(n+10)!]^{3}(4n+7)!!}]}[/tex]
[tex]\frac{a_{n}}{a_{n+1}}=\frac{(n+10)^{3}(4n+7)(4n+5)}{32(n+1)(3n+4)(3n+5)(3n+6)+(2n+2)}[/tex]
Applying n → ∞
[tex]\lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}= \lim_{n \to \infty}\frac{(n+10)^{3}(4n+7)(4n+5)}{32(n+1)(3n+4)(3n+5)(3n+6)+(2n+2)}[/tex]
The numerator as well denominator of [tex]\frac{a_{n}}{a_{n+1}}[/tex] are polynomials of fifth degree with leading coefficients:
[tex](1^{3})(4)(4)=16\\(32)(1)(3)(3)(3)(2)=1728\\ \lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}=\frac{16}{1728}=\frac{1}{108}[/tex]
What is the selling price of merchandise listed at $5,900 if discounts of 15%, 10%, and 4% are given?
Answers
Answer:
after 15%, discount = $5,015
after 10%, discount = $5,310
after 4%, discount= $5,664
Step-by-step explanation:
1.discount of 15 % of the price listed
so 15 % of $5,900 will be= 15/100 x 5900 = 885 $
so after discounting the price =$5,900 - $885 = $5,015
2. discount of 10 % of the price listed
so 10 % of $5,900 will be= 10/100 x 5900 = 590 $
so after discounting the price =$5,900 - $590 = $5,310
3. discount of 4 % of the price listed
so 4 % of $5,900 will be= 4/100 x 5900 = 236 $
so after discounting the price =$5,900 - $236 = $5,664
A rancher purchased an SUV for $33,714 and made a down payment of 15% of the cost. The balance was financed for 4 years at an annual interest rate of 7%. Find the monthly truck payment.
Answers
Formula for monthly payment is:
A = P x (r(1+r)^t)/((1+r)^t-1) where P is the amount financed, r is the interest rate divided by 12 and t is the amount of time for the loan in months.
P = 33714 x 0.85 = 28656.90
A = 28656.90 x (0.07/12 (1+0.07/12)^48) / (1 +0.07/12)^48 - 1)
A = $686.23
Conduct a test at the a=0.05 level of significance by determining (a) null and alternative hypothesis, (b) the test statistic, (c) the P-value. Assume the samples were obtained independently from a large population using simple random sampling. Test whether p1 > p2. The sample data are x1=117 n1=249 x2=141 n2=312
Answers
Answer:
Since p > alpha, we accept H0, there is no evidence to prove that p1 is greater than p2
Step-by-step explanation:
Set up hypotheses as
[tex]H_0: p1 = p2\\H_a: p1>p2[/tex]
(Right tailed t test)
Alpha = 0.05
Sample I II Total
X 117 141 248
N 249 312 561
p 0.4700 0.4519 0.4420
p difference = 0.0181
Std dev = [tex]\sqrt{p(1-p)(\frac{1}{n_1} +\frac{1}{n_2} )}[/tex]
=0.0427
z statistic = 0.424
p value = 0.33724
Since p > alpha, we accept H0, there is no evidence to prove that p1 is greater than p2
Final answer:
To test whether p1 is greater than p2 using hypothesis testing, one needs to state the null and alternative hypotheses, calculate the test statistic and p-value, and then decide whether to reject or accept the null hypothesis based on the level of significance and the p-value. Calculating the test statistic and p-value requires specific formulas and is not provided here.
Explanation:
Conducting a Hypothesis Test for Two Population Proportions
A student would like assistance in conducting a hypothesis test for two population proportions. The steps to perform such a test include:
State the null and alternative hypotheses: The null hypothesis (H0) is that there is no difference between the two population proportions (p1 - p2 = 0), while the alternative hypothesis (Ha) posits that there is a difference (p1 > p2).
The random variable (P') represents the difference between the two sample proportions.
Calculate the test statistic: Using the provided sample sizes and successful outcomes, the test statistic is calculated using a formula based on the Z-distribution.
Calculate the p-value: The p-value is determined from the test statistic, indicating the probability of observing such a result if the null hypothesis were true.
At the 5 percent level of significance, compare the p-value to the alpha value of 0.05 to make a decision. If the p-value is less than alpha, reject the null hypothesis.
The Type I error would occur if the null hypothesis is incorrectly rejected when it is actually true.
The Type II error would occur if the null hypothesis is not rejected when it is actually false.
The test statistic and p-value cannot be calculated from the information provided without the appropriate formulas or statistical tools.
Decision Making and Errors
If the alpha level is greater than the p-value, the null hypothesis should be rejected, indicating there is evidence to suggest p1 is greater than p2.
Failure to reject the null hypothesis when it is false constitutes a Type II error.
simplify (3/4 + 4/5i)-(1/2 - 3/10i)
Answers
For this case we must simplify the following expression:
[tex](\frac {3} {4} + \frac {4} {5} i) - (\frac {1} {2} - \frac {3} {10} i) =[/tex]
By law of multiplication signs we have to:
[tex]- * + = -\\- * - = +\\\frac {3} {4} + \frac {4} {5} i- \frac {1} {2} + \frac {3} {10} i =[/tex]
We add similar terms:
[tex]\frac {3} {4} - \frac {1} {2} + \frac {4} {5} i + \frac {3} {10} i =\\\frac {3 * 2-4 * 1} {4 * 2} + \frac {4 * 10 + 5 * 3} {10 * 5} i =\\\frac {2} {8} + \frac {55} {50} i =[/tex]
We simplify:
[tex]\frac{1}{4}+\frac{11}{10}i[/tex]
Answer:
[tex]\frac{1}{4}+\frac{11}{10}i[/tex]