Answer: The Sun is the major source of energy for organisms and the ecosystems of which they are a part. Producers such as plants, algae, and cyanobacteria use the energy from sunlight to make organic matter from carbon dioxide and water. This establishes the beginning of energy flow through almost all food webs.
Hope you understood : )Question no 6 answer
Answer:
a = 1b = -28P(x) = (x +2)(x -3)(x^2 +3x +10)Step-by-step explanation:
a) Since x+2 is a factor, we know P(-2) = 0.
P(x) = (((x +2)x +a)x +b)x -60
Then the value of P(-2) is ...
P(-2) = 0 = (((0)(-2) +a)(-2) +b)(-2) -60 = (-2a +b)(-2) -60 = 4a -2b -60
We know the remainder from division by (x+3) is 60, so
P(-3) = 60 = (((-3+2)(-3) +a)(-3) +b)(-3) -60 = ((3+a)(-3) +b)(-3) -60
= (-9 -3a +b)(-3) -60 = 27 +9a -3b -60
93 = 9a -3b
These two equations can be put into standard form:
2a -b = 30
3a -b = 31
Then we have the solution ...
a = 1 . . . . . (by subtracting the first equation from the second)
-28 = b . . . by substituting into the first equation
__
b) To show that (x-3) is a factor we need to evaluate P(3).
P(3) = (((3 +2)(3) +1)(3) -28)(3) -60 = (48 -28)(3) -60 = 0
The function value is 0, so (x -3) is a factor.
__
c) We want to find Q(x) = x^2 +cx +d such that ...
(x +2)(x -3)Q(x) = P(x)
(x^2 -x -6)(x^2 +cx +d) = x^4 +2x^3 +x^2 -28x -60
x^4 +(c-1)x^3 +(-6-c+d)x^2 +(-6c-d)x -6d = x^4 +2x^3 +x^2 -28x -60
This gives rise to the equations ...
c -1 = 2 ⇒ c = 3
-6d = -60 ⇒ d = 10
Then P(x) can be factored as ...
P(x) = (x +2)(x -3)(x^2 +3x +10)
_____
Comment on the attached graph
I like to use a graphing calculator to find real roots of higher-degree polynomials. This graph shows the real zeros to be -2 and +3, so we know that (x +2) and (x -3) are factors. The green curve is P(x) with those factors divided out, so is a graph of Q(x). The vertex of that graph tells us that ...
Q(x) = (x +1.5)^2 +7.75 = x^2 +3x +10
Describe the key features of polynomial function
g(x) = x^3 − x^2 − 4x + 4, including the end behavior, y-intercept, and zeros.
Answer:
Zeros: x = 1 , − 2 , 2 End Behavior: Falls to the left and rises to the right.
Y-intercept: (0,4)
Step-by-step explanation:
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Suppose that you are interested in the average GPA for all KCTCS students. In order to conduct the study you take a sample of 1500 students. This is an example of what branch of statistics?
Inferential
Descriptive
Inductive
Deductive
Answer:
Descriptive
Step-by-step explanation:
Descriptive statistics uses the data that helps to analyse, describe, and elaborate data accurately. It also provides information regarding the population through graphs or tables. Four types of descriptive statistics are measures of frequency, measures of central tendency, measures of dispersion, and measures of position.
The given example in question is descriptive statistics.
Someone help please is been so long seems I don’t do this
Answer:
0.5 or 1/2
Step-by-step explanation:
Step 1. 40/50−15/50
Step 2. 40-25= 25/50
Step 3. 25/50÷ 5 =5/10
5/10÷ 5 =1/2
4/5−3/10=1/2
Nancy purchased shares of Micron for $51.33 per share. She plans to sell the shares when the stock price rises 35%. At what price will she sell her shares? *
Answer: $69.30 per share (rounded up to the nearest penny)
Step-by-step explanation:
51.33 x 0.35 = 17.9655 (this is the increased value in the stock)
51.33 + 17.9655 = 69.2955
There is a cuboids box with the
length 4, width 2 and height 3. Now
I will have to calculate the area of
the box. So what will be the area?
A random survey of 1000 students nationwide showed a mean ACT score of 21.1. Ohio was not used. A survey of 500 randomly selected Ohio scores showed a mean of 20.8. The population standard deviation is 3. The goal of the study is to decide if we can conclude that Ohio is below the national average. Use α = 0.1. What would be the hypotheses. Identify the claim. What would be the critical value(s).
Answer:
We conclude that the mean Ohio score is below the national average.
Step-by-step explanation:
We are given that a random survey of 1000 students nationwide showed a mean ACT score of 21.1. Ohio was not used.
A survey of 500 randomly selected Ohio scores showed a mean of 20.8. The population standard deviation is 3.
Let [tex]\mu[/tex] = mean Ohio scores.
So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] [tex]\geq[/tex] 21.1 {means that the mean Ohio score is above or equal the national average}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 21.1 {means that the mean Ohio score is below the national average}
The test statistics that would be used here One-sample z test statistics as we know about population standard deviation;
T.S. = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex] ~ N(0,1)
where, [tex]\bar X[/tex] = sample mean Ohio score = 20.8
[tex]\sigma[/tex] = population standard deviation = 3
n = sample of Ohio = 500
So, test statistics = [tex]\frac{20.8-21.1}{\frac{3}{\sqrt{500}}}[/tex]
= -2.24
The value of z test statistics is -2.24.
Now, at 0.1 significance level the z table gives critical value of -1.2816 for left-tailed test. Since our test statistics is less than the critical values of z as -2.24 < 1.2816, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.
Therefore, we conclude that the mean Ohio score is below the national average.
The hypotheses for the given survey are; Null Hypothesis; H₀: μ₁ = μ₂
Alternative Hypothesis; Hₐ: μ₁ > μ₂ (claim)
What is the hypotheses?
We are given;
Population Size; N = 1000
Sample size; n = 500
Population mean; μ = 21.1
Sample Mean; x' = 20.8
Population Standard Deviation; σ = 3
Let us first define the hypotheses;
Null Hypothesis; H₀: μ₁ = μ₂
Alternative Hypothesis; Hₐ: μ₁ > μ₂
Since the population mean was not used, then it means that the Alternative Hypothesis is the claim.
We are told that significance value is α = 0.1. Using F-distribution table attached with; α = 0.1; dF₁ = 20 and dF₂ = 20, we have;
Critical Value = 1.79
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Suppose one of Miguel’s scores from the Water Hole is selected at random. What is the probability that Miguel’s score on the Water Hole is at most 5 ?
Complete Question
The complete question is shown on the first uploaded image
Answer:
The probability is [tex]P(X < 5) = 0.80[/tex]
Step-by-step explanation:
The probability that Miguel’s score on the Water Hole is at most 5 is mathematically evaluated as
[tex]P(X < 5) = P(X = 3) + P(X = 4 ) + P(X = 5)[/tex]
Substituting values from the table
[tex]P(X < 5) = 0.15 + 0.40 + 0.25[/tex]
[tex]P(X < 5) = 0.80[/tex]
Which inequality is true? Use the number line to help.
Answer:
C) -1.5 < -0.5
Question 7 ( 6 Marks)
Three crayons are held together with an elastic band. The diagram below shows the end of
the crayons and the elastic band. Each of the crayons has diameter of 10 mm. Find the
length of the elastic band in this position.
Answer:
Total length of the elastic band in this stretched position = 61.43 mm
Step-by-step explanation:
The complete Question with the image is attached to this solution
The second image shows the break down of the angle subtended by the elastic band over the crayons at the centre of the circle formed by the crayon's circular shape.
The elastic band extends over 3 of such length of an arc and covers a length of 2 × radius in the spaces between the contact with each crayon. This is also shown in the second attached image.
So, the length of the band will be a sum of 3 diameters and 3 of those arc lengths
Length of an arc is given by
L = (θ/360) × πD
θ = 120°
D = 10 mm
L = (1/3) × 10π = (10π/3) mm
But there are 3 of those, So, total length of the arcs = 3 × (10π/3) = (10π) mm
Length of those 3 diameter regions = 3×10 = 30 mm
Total length of the elastic band in this stretched position = 30 + 10π = 30 + 31.43 = 61.43 mm
Hope this Helps!!!
The solution to the system of equation
-8x+4y=24
Answer:
x-intercept=(-3,0) y-intercept= (0,6)
Step-by-step explanation:
Hopefully this is the answer you wanted if not please let me know by commenting
Write a polynomial function that meets the given conditions. Answers may vary. Degree 2 polynomial with zeros of 2√ 13 and -2√13
Answer:
The polynomial function is [tex]x^{2} - 52[/tex]
Step-by-step explanation:
A polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = (x - x_{1})*(x - x_{2})[/tex].
In this problem:
The roots are [tex]x_{1} = 2\sqrt{13}[/tex] and [tex]x_{2} = -2\sqrt{13}[/tex]
Then
[tex](x - 2\sqrt{13}) \times (x - (-2\sqrt{13})) = (x - 2\sqrt{13}) \times (x + 2\sqrt{13}) = x^{2} - 2x\sqrt{13} + 2x\sqrt{13} -(2\sqrt{13})^{2} = x^{2} - 52[/tex]
The polynomial function is [tex]x^{2} - 52[/tex]
1.
Write a polynomial function in factored form with zeros at -2, 5, and 6.
Answer:
f(x) = (x + 2) (x − 5) (x − 6)
Step-by-step explanation:
f(x) = (x − (-2)) (x − 5) (x − 6)
f(x) = (x + 2) (x − 5) (x − 6)
Which of these shows line of symmetry?
Select the three correct answers.
A researcher wants to check the claim that convicted burglars spend an average of 18.7 months in jail. She takes a random sample of 11 such cases from court files and finds that the data was normally distributed and average jail time was 15.5 months with a standard deviation of 5.7 months. Is the reported claim correct? (use a 0.05 level of significance)
Answer:
[tex]t=\frac{15.5-18.7}{\frac{5.7}{\sqrt{11}}}=-1.862[/tex]
The degrees of freedom are given by:
[tex]df=n-1=11-1=10[/tex]
The p value would be given by:
[tex]p_v =2*P(t_{(10)}<-1.862)=0.0922[/tex]
We see that the p value is higher than the significance level so we don't have enough evidence to ocnclude that the true mean is different from 18.7 months in jail at 5% of significance.
Step-by-step explanation:
Information given
[tex]\bar X=15.5[/tex] represent the sample mean for the jail time
[tex]s=5.7[/tex] represent the sample standard deviation
[tex]n=11[/tex] sample size
[tex]\mu_o =18.7[/tex] represent the value that we want to compare
[tex]\alpha=0.05[/tex] represent the significance level
t would represent the statistic
[tex]p_v[/tex] represent the p value for the test
System of hypothesis
We want to check the hypothesis if the true mean for the jail time is equal to 18.7 or no, the system of hypothesis are:
Null hypothesis:[tex]\mu = 18.7[/tex]
Alternative hypothesis:[tex]\mu \neq 18.7[/tex]
Since we don't know the population deviation the statistic is given by:
[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)
Rreplacing we got:
[tex]t=\frac{15.5-18.7}{\frac{5.7}{\sqrt{11}}}=-1.862[/tex]
The degrees of freedom are given by:
[tex]df=n-1=11-1=10[/tex]
The p value would be given by:
[tex]p_v =2*P(t_{(10)}<-1.862)=0.0922[/tex]
We see that the p value is higher than the significance level so we don't have enough evidence to ocnclude that the true mean is different from 18.7 months in jail at 5% of significance.
The function rule y = –0.5cos(x) – 3 describes graph shown.
answer is c
Answer:
The answer is C.
Step-by-step explanation:
It is down 3 and it is not D. Just got it right too.
Answer:
Answer is C on Edge 2020
Below is a simplified version of a model developed to investigate the impact of stock
splits on long-run stock performance for acquiring firms:
y = Bo+ Bıx1+B2x2 + B3x1x2
Where: y = average salary; x1 = {1, if stock split prior to acquisition, 0 if not } x2 = {1 if
firm's discretionary accrual is high, 0 if discretionary accrual is low.
A) In terms of the B's in the model, what is the mean buy-out-and-hold return rate (BAR)
for a firm with no stock split and a high discretionary accrual?
B) In terms of the B's in the model, what is the mean buy-out-and-hold return rate (BAR)
for a firm with no stock split and a low discretionary accrual?
C) For firms with no stock split, find the difference between the mean BAR for firms
with high and low DA. (Hint: Use your answers to parts a and b).
D) For firms with a stock split, find the difference between the mean BAR for firms with
high and low DA. (Hint: Use your answers to parts a and b)
Answer :
(a).When
x1 = 0, x2 = 1
y = b0 + b2
(b).x1 = 0 , x2 = 0
y= b0
(c). Difference = (b0 + b2) - b0 = b2
(d). when stock split
x1 = 1
Hence difference = (b0 + b1 + b2 + b3) - (b0 + b1)
= (b2 + b3)
The base of a rectangular box measures 3 feet by 4 feet. What is the height in feet of the box if the volume is 48 cubic feet
Answer:
3 * 4 * height = 48
height = 48 / (3*4)
height = 4 feet
Step-by-step explanation:
The height is 4 feet of the box if the volume is 48 cubic feet.
What is the volume of the box?The volume of a rectangular box can be calculated if you know its three dimensions: width, length, and height.
The formula is then volumebox = width x length x height.
The base of a rectangular box measures 3 feet by 4 feet.
The volume is 48 cubic feet.
Substitute all the values in the formula
Volume box = width x length x height
48 = 3 x 4 x height
48 = 12 x height
Height = 48/12
Height = 4
Hence, the height is 4 feet of the box if the volume is 48 cubic feet.
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Use the Law of Sines to solve the triangle. (Let b = 47.7 yd. Round your answers for a and c to two decimal places.)
Answer:
C = 68.667°
a = 123.31 yd.
c = 114.90 yd.
Step-by-step explanation:
The missing image for the question is attached to this solution.
In the missing image, a triangle AB is given with angles A and B given to be 88° 35' and 22° 45' respectively
We are them told to find angle C and side a and c given that side b = 47.7 yd.
A = 88° 35' = 88° + (35/60)° = 88.583°
B = 22° 45' = 22° + (45/60)° = 22.75°
The sum of angles in a triangle = 180°
A + B + C = 180°
C = 180° - (A + B) = 180° - (88.583° + 22.75°) = 68.667°
The sine law is given as
(a/sin A) = (b/sin B) = (c/sin C)
Using the first two terms of the sine law
(a/sin A) = (b/sin B)
a = ?
A = 88.583°
b = 47.7 yd.
B = 22.75°
(a/sin 88.583°) = (47.7/sin 22.75°)
a = (47.7 × sin 88.583°) ÷ sin 22.75°
a = 123.31 yd.
Using the last two terms of the sine law
(b/sin B) = (c/sin C)
b = 47.7 yd.
B = 22.75°
c = ?
C = 68.667°
(47.7/sin 22.75°) = (c/sin 68.667°)
c = (47.7 × sin 68.667°) ÷ sin 22.75°
c = 114.90 yd.
Hope this Helps!!!
Dolores bought a crib on sale for $350. The sale price was 40% of the original price, what was the original price?
Step-by-step explanation:
Hello there!
Just divide 350 by 40, and find the full percentage by multiplying it by 100.
$350/40=8.75
$8.75*100=$875
:)
The original price of the crib was $875.
To find the original price of the crib, use the information given.
Let's assume the original price of the crib is represented by x.
According to the given information, the sale price of $350 is 40% of the original price.
40% of x = $350
To calculate 40% of x, convert the percentage to decimal form by dividing it by 100:
(40/100) × x = $350
Simplifying:
0.4x = $350
To solve for x, divide both sides of the equation by 0.4:
x = $350 / 0.4
x = $875
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Answer I need help don’t get it
Answer:
x ↑2 + ( y − 1 )↑ 2 = 16
Step-by-step explanation:
Standard form.
Mark me as brainliest please!
In designing a true/false test you use a random number generator to determine if the answer is true or false. This gives you a run of 5 true answers in a row in a test of 20 questions. You think this is unusual so you conduct a simulation to test this. The number of times a run of 5 trues appears in each set of 20 is what you record. You do this 50 times. Are your results unusual? Explain.
Answer:
no because
Step-by-step explanation:
the results were not tested enough
Answer:
The number of times a run of 5 trues appears in each set of 20 is what you record. You do this 50 times. Are your results unusual? Explain.
C) Their results were typical. Almost half of the simulations had runs of five or more true answers.
Step-by-step explanation:
100 points plz answer.....What is the volume of the prism?
8m
6m
14m
Answer:
672m
Step-by-step explanation:
L*W*H
8*6*14
672m
On January 1, 2021, White Water issues $410,000 of 7% bonds, due in 10 years, with interest payable semiannually on June 30 and December 31 each year.
Assuming the market interest rate on the issue date is 8%, the bonds will issue at $382,141.
Record the bond issue on January 1, 2021, and the first two semiannual interest payments on June 30, 2021, and December 31, 2021. (If no entry is required for a particular transaction/event, select "No Journal Entry Required" in the first account field. Round your final answers to the nearest whole dollar.)
Answer:
Step-by-step explanation:
The appropriate journal entries to record the bond issue on January 1, 2021, and the first two semiannual interest payments on June 30, 2021, and December 31, 2021 are:
White Water journal entries
1-Jan-21
Debit Cash $382,141
Credit Discount on Bonds Payable $27,859
($410,000-$382,141)
Credit Bonds payable $ 410,000
30-Jun
Debit Interest Expenses $ 15,286
($382,141 x 8%/2)
Debit Discount on Bonds Payable $736
Credit Cash $14,350
($410,000 x 7%/2)
31-Dec
Debit Interest Expenses $15,315.08
[($382,141 + 736) x 8%/2]
Credit Discount on Bonds Payable $965.08
($15,315.08-$14,350)
Credit Cash $14,350
($410,000 x 7%/2)
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write zeros in the divided 49÷14
Answer:
3.50
Step-by-step explanation:
The attached shows the long division with zeros filled where necessary.
100 POINTS
PLEASE PROVIDE STEPS AND GRAPH.
GRAPH USING DESMOS IF POSSIBLE.
Answer:
A) (-2, 8), (4, 8)
B) See attachment.
C) 36 square units
Step-by-step explanation:
Part AGiven functions:
[tex]\begin{cases}f(x)=x^2-2x\\g(x)=8\end{cases}[/tex]
To find the x-coordinates of the points where the two functions intersect, set the equations equal to each other and solve for x:
[tex]f(x)=g(x)\\\\x^2-2x=8\\\\x^2-2x-8=0\\\\x^2-4x+2x-8=0\\\\x(x-4)+2(x-4)=0\\\\(x+2)(x-4)=0\\\\\\x+2=0 \implies x=-2\\\\x-4=0 \implies x=4[/tex]
As points on the function g(x) have a y-coordinate of 8, the solutions to f(x) = g(x) are:
[tex]\Large\boxed{(-2,8)\;\textsf{and}\;(4,8)}[/tex]
[tex]\dotfill[/tex]
Part BThe function g(x) is a horizontal line parallel to the x-axis that intercepts the y-axis at (0, 8).
We are given that function f(x) is a concave up parabola with roots at (0, 0) and (2, 0). From Part A, we know that the parabola intersects g(x) at points (-2, 8) and (4, 8).
To help sketch the graph of function f(x), we can find its vertex and y-intercept.
The x-coordinate of the vertex of a quadratic function in the form y = ax² + bx + c is x = -b/2a. In this case, a = 1 and b = -2, so the x-coordinate of the vertex is:
[tex]x = -\dfrac{-2}{2(1)}=\dfrac{2}{2}=1[/tex]
To find the y-coordinate of the vertex, substitute x = 1 into f(x):
[tex]f(1)=(1)^2-2(1)=1-2=-1[/tex]
Therefore, the vertex of f(x) is at point (1, -1). The axis of symmetry of a parabola is the x-coordinate of the vertex, so in this case, the axis of symmetry is x = 1.
To find the y-intercept, substitute x = 0 into the function and solve for y:
[tex]f(0)=(0)^2-2(0)=0-0=0[/tex]
Therefore, the y-intercept of f(x) is the origin (0, 0).
[tex]\dotfill[/tex]
Part CThe area of the region R bounded by f(x) and g(x) on the interval [-2, 4] is given by the integral:
[tex]\displaystyle R = \int_{-2}^{4} \left[ g(x) - f(x) \right] \; dx[/tex]
Substitute g(x) = 8 and f(x) = x² - 2x:
[tex]\displaystyle R = \int_{-2}^{4} \left[8-(x^2-2x)\right] \; dx \\\\\\ R = \int_{-2}^{4} \left[8-x^2+2x\right] \; dx[/tex]
Evaluate the Integral
[tex]\displaystyle R = \left[8x-\dfrac{x^{2+1}}{2+1}+\dfrac{2x^{1+1}}{1+1}\right]^4_{-2} \\\\\\ R = \left[8x-\dfrac{x^{3}}{3}+x^2\right]^4_{-2} \\\\\\ R=\left(8(4)-\dfrac{(4)^{3}}{3}+(4)^2\right)-\left(8(-2)-\dfrac{(-2)^{3}}{3}+(-2)^2\right) \\\\\\ R=\left(32-\dfrac{64}{3}+16\right)-\left(-16+\dfrac{8}{3}+4\right) \\\\\\ R=\dfrac{80}{3}+\dfrac{28}{3} \\\\\\ R=\dfrac{108}{3}\\\\\\ R=36[/tex]
Therefore, the area of the region R bounded by f(x) and g(x) is:
[tex]\Large\boxed{\textsf{$36$ square units}}[/tex]
hilip Morris wishes to determine if there is a difference between the proportion of women and proportion of men who smoke cigarettes. Random samples of 125 women and 140 men reveal that 13 women and 5 men smoke cigarettes. Does the data indicate that the proportion of women who smoke cigarettes is higher than the proportion of men who do at α=.01? What is the 95% confidence interval for pWomen - pMen?
Answer:
We conclude that the proportion of women who smoke cigarettes is smaller than or equal to the proportion of men at 0.01 significance level.
95% confidence interval for the difference in population proportions of women and men who smoke cigarettes is [0.0062 , 0.1298].
Step-by-step explanation:
We are given that random samples of 125 women and 140 men reveal that 13 women and 5 men smoke cigarettes.
Let [tex]p_1[/tex] = population proportion of women who smoke cigarettes
[tex]p_2[/tex] = population proportion of men who smoke cigarettes
So, Null Hypothesis, [tex]H_0[/tex] : [tex]p_1\leq p_2[/tex] {means that the proportion of women who smoke cigarettes is smaller than or equal to the proportion of men}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]p_1> p_2[/tex] {means that the proportion of women who smoke cigarettes is higher than the proportion of men}
The test statistics that will be used here is Two-sample z proportion test statistics;
T.S. = [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] ~ N(0,1)
where, [tex]\hat p_1[/tex] = sample proportion of women who smoke cigarettes= [tex]\frac{13}{125}[/tex] =0.104
[tex]\hat p_2[/tex] = sample proportion of men who smoke cigarettes = [tex]\frac{5}{140}[/tex] = 0.036
[tex]n_1[/tex] = sample of women = 125
[tex]n_2[/tex] = sample of men = 140
So, the test statistics = [tex]\frac{(0.104-0.036)-(0)}{\sqrt{\frac{0.104(1-0.104)}{125}+ \frac{0.036(1-0.036)}{140}} }[/tex]
= 2.158
Now, at 0.01 significance level, the z table gives critical value of 2.3263 for right tailed test. Since our test statistics is less than the critical value of z as 2.158 < 2.3263, so we have insufficient evidence to reject our null hypothesis due to which we fail to reject our null hypothesis.
Therefore, we conclude that the proportion of women who smoke cigarettes is smaller than or equal to the proportion of men.
Now, coming to 95% confidence interval;
Firstly, the pivotal quantity for 95% confidence interval for the difference between population proportions is given by;
P.Q. = [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] ~ N(0,1)
where, [tex]\hat p_1[/tex] = sample proportion of women who smoke cigarettes= [tex]\frac{13}{125}[/tex] =0.104
[tex]\hat p_2[/tex] = sample proportion of men who smoke cigarettes = [tex]\frac{5}{140}[/tex] = 0.036
[tex]n_1[/tex] = sample of women = 125
[tex]n_2[/tex] = sample of men = 140
Here for constructing 95% confidence interval we have used Two-sample z proportion statistics.
So, 95% confidence interval for the difference between population proportions, [tex](p_1-p_2)}[/tex] is ;
P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level
of significance are -1.96 & 1.96}
P(-1.96 < [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] < 1.96) = 0.95
P( [tex]-1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] < [tex]{(\hat p_1-\hat p_2)-(p_1-p_2)}[/tex] < [tex]1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] ) = 0.95
P( [tex](\hat p_1-\hat p_2)-1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] < [tex](p_1-p_2)}[/tex] < [tex](\hat p_1-\hat p_2)+1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex] ) = 0.95
95% confidence interval for [tex](p_1-p_2)}[/tex] =
[[tex](\hat p_1-\hat p_2)-1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex],[tex](\hat p_1-\hat p_2)+1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex]]
= [ [tex](0.104-0.036)-1.96 \times {\sqrt{\frac{0.104(1-0.104)}{125}+ \frac{0.036(1-0.036)}{140}} }[/tex] , [tex](0.104-0.036)+1.96 \times {\sqrt{\frac{0.104(1-0.104)}{125}+ \frac{0.036(1-0.036)}{140}} }[/tex] ]
= [0.0062 , 0.1298]
Therefore, 95% confidence interval for the difference in population proportions of women and men who smoke cigarettes is [0.0062 , 0.1298].
There is growing concern over a possible causal link between consumption of diet soda and obesity. In 2013, Gallup took a nationally representative sample of 2027 adults and asked them about their soda consumption. In the sample, 920 adults described themselves as overweight, and 300 of them said that they mostly drink diet soda. The remaining 1107 adults described themselves as about right weight, and 225 of them said that they mostly drink diet soda. We plan to test the following hypotheses: The pooled sample proportion of individuals who mostly drink diet soda is: a. 0.2491 b. 0.2033 c. 0.5000 d. 0.2590 e. 0.3261
Answer:
d. 0.2590
Step-by-step explanation:
In this problem, we have that:
A sample of 2027 adults.
920 overweight, and of those, 300 said they drink mostly diet soda.
1107 right weight, of those 225 said that they mostly drink diet soda.
Of 2027 adults, 300+225 = 525 said that they mostly drink diet soda.
The pooled sample proportion of individuals who mostly drink diet soda is:
525 of 2027
525/2027 = 0.2590
So the correct answer is:
d. 0.2590
Binomial Distribution Problem 1: (a) An urn contains 1000 balls, 100 are green and 900 are white. One ball is chosen from the urn 100 times with replacement. Use Excel (binom.dist) to find the probability that six or seven green balls are selected. (b) An urn contains 1000 balls, 100 are green and 900 are white. One ball is chosen from the urn 1000 times. Use Excel (binom.dist) to find the probability that between 110 and 120 of the balls, inclusive, are green. (c) Redo (a) and (b) again using Excel but use the normal approximation (normal.dist). How do the answers compare with the above? Are there any discrepancies? If so, please explain why they happened. Please submit your answers on an excel spreadsheet.
Find the attachments for solution and explanation
The student's question regards calculating binomial probabilities and normal approximations using Excel. Precise probabilities are found using the BINOM.DIST function, and normal approximations are made with NORM.DIST. Discrepancies can arise due to the approximation not perfectly representing the discrete binomial outcomes.
In solving problems using binomial probabilities with Excel, the function =BINOM.DIST(number_s, trials, probability_s, cumulative) is used to calculate the probability of a specified number of successes in a series of independent trials. In problem (a), to find the probability that six or seven green balls are selected, we would use =BINOM.DIST(6, 100, 0.1, FALSE) and =BINOM.DIST(7, 100, 0.1, FALSE) adding both probabilities together. For problem (b), to find the probability that between 110 and 120 green balls are chosen, we calculate the cumulative probability for 120 and subtract the cumulative probability for 109 using =BINOM.DIST(120, 1000, 0.1, TRUE) - BINOM.DIST(109, 1000, 0.1, TRUE).
The normal approximation can be applied to the binomial distribution when the number of trials is large and the success probability is not too close to 0 or 1, using Excel's =NORM.DIST(x, mean, standard_dev, cumulative) function. Comparing the results of the normal approximation with the exact binomial probabilities may reveal discrepancies due to the approximation being less accurate for probabilities that are far from the mean, especially when the success probability (p) is not near 0.5, or when the number of trials (n) is not large enough. These discrepancies are due to the smooth curve assumption in the normal distribution approximation, which may not perfectly represent the discrete nature of binomial outcomes.
Bin $A$ has one white ball and four black balls. Bin $B$ has three balls labeled $\$1$ and one ball labeled $\$7$. Bin $W$ has five balls labeled $\$8$ and one ball labeled $\$500$. A game is played as follows: a ball is randomly selected from bin $A$. If it is black, then a ball is randomly selected from bin $B$; otherwise, if the original ball is white, then a ball is randomly selected from bin $W$. You win the amount printed on the second ball selected. What is your expected win
Answer:
$20
Step-by-step explanation:
Bin A: Bin B: Bin W:
white ball: 20% $1: 75% $8: 5/6
black ball: 80% $7: 25% $500: 1/6
first we can calculate the expected return of bins B and W:
expected return if the ball is black = ($1 x 75%) + ($7 x 25%) = $2.50
expected return if the ball is white = ($8 x 5/6) + ($500 x 1/6) = $90
expected return of the game = (expected return of a black ball x probability of choosing a black ball) + (expected return of a white ball x probability of choosing a white ball) =($2.50 x 80%) + ($90 x 20%) = $2 + $18 = $20
Answer:
$20
Step-by-step explanation:
Since Bin A has one white ball and four black balls, the money ball has a 1/5 chance of coming from Bin W and a 4/5 chance of coming from Bin B. The total expected value therefore is $E = 1/5E_W+4/5E_B, where E_W and E_B are the expected values of a ball drawn from bins W and B, respectively. Since Bin W has five 8 dollar balls and one 500 dollar ball, its expected value is E_W = 5/6*8 + 1/6*500 = 90. Since Bin B has three 1 dollar balls and one 7 dollar ball, its expected value is E_B = 3/4*1 + 1/4*7 = $2.5. Therefore E = 1/5E_W + 4/5E_B = 1/5*90 + 4/5*2.5 = 20