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Statistics, section 2, Fall 2019
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Wackerly et al Mathematical Stats Applications 7/e (Homework)

James Finch

Statistics, section 2, Fall 2019

Instructor: Dr. Friendly

Current Score : 20 / 63

Due : Sunday, January 27, 2030 23:30 EST

Last Saved : n/a Saving... ()

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1	2	3	4	5	6	7	8	9	10	11
3/3	3/4	6/6	7/7	1/10	0/3	0/4	–/4	–/4	–/8	–/10

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20/63 (31.7%)

Instructions

In their bestselling book, Mathematical Statistics With Applications, 7th edition, published by Cengage Learning, premiere authors Dennis Wackerly, William Mendenhall, and Richard L. Scheaffer, present a solid foundation in statistical theory while conveying the relevance and importance of the theory in solving practical problems in the real world. The authors' use of practical applications and excellent exercises helps students discover the nature of statistics and understand its essential role in scientific research.

Question 1 demonstrates interpreting relative frequency histograms.

Question 2 demonstrates creating a sample space and entering it as a comma separated list.

Question 3 demonstrates defining and graphing a discrete probability distribution and using it to find a specific probability.

Question 4 demonstrates finding and graphing the cumulative distribution function given a density function.

Question 5 demonstrates working with joint density functions.

Question 6 demonstrates entry of a symbolic answer.

Question 7 demonstrates calculating and interpreting a confidence interval. Links to tables and technology are provided.

Question 8 demonstrates how limits can be evaluated in a step-by-step fashion.

New for Fall 2022! Question 9 demonstrates fitting a quadratic model using R.

New for Fall 2022! Question 10 demonstrates using R to analyze a matched pairs experiment.

New for Fall 2022! Question 11 demonstrates using R for an Analysis of Variance for a Randomized Block Design. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

The answer key and solutions will display after the first submission for demonstration purposes. Instructors can configure these to display after the due date or after a specified number of submissions.

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. 3/3 points | Previous Answers WackerlyStat7 1.2.005. My Notes

Given below is the relative frequency histogram associated with grade point averages (GPAs) of a sample of 30 students.

(a)

Which of the GPA categories identified on the horizontal axis are associated with the largest proportion of students? (Select all that apply.)

(b)

What proportion of students had GPAs in each of the categories that you identified?

(c)

What proportion of the students had GPAs less than 2.45?

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2. 3/4 points | Previous Answers WackerlyStat7 2.5.027. My Notes

Consider a situation where cars entering an intersection each could turn right, turn left, or go straight. An experiment consists of observing two vehicles moving through the intersection.

(a)

How many sample points are there in the sample space?

List the events in the sample space. (Enter your answer in set notation. Use G for go straight, L for left, and R for right.)

S =

{GG,GL,GR,LG,LL,LR,RG,RL,RR}

(b)

Assuming that all sample points are equally likely, what is the probability that at least one car turns left? (Enter your probability as a fraction.)

(c)

Again assuming equally likely sample points, what is the probability that at most one vehicle turns? (Enter your probability as a fraction.)

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3. 6/6 points | Previous Answers WackerlyStat7 3.2.009. My Notes

In order to verify the accuracy of their financial accounts, companies use auditors on a regular basis to verify accounting entries. The company's employees make erroneous entries 22% of the time. Suppose that an auditor randomly checks three entries.

(a)

Find the probability distribution for Y, the number of errors detected by the auditor.

P(Y = 0) =

P(Y = 1) =

P(Y = 2) =

P(Y = 3) =

(b)

Construct a probability histogram for

p(y).

(c)

Find the probability that the auditor will detect more than one error.

P(Y > 1) =

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4. 7/7 points | Previous Answers WackerlyStat7 4.2.011. My Notes

Suppose that Y possesses the density function

f(y) =

cy,	0 ≤ y ≤ 2,
0,	elsewhere.

(a)

Find the value of c that makes

f(y)

a probability density function.

c =

(b)

Find

F(y).

F(y) =

0 ,	y < 0
14`y`2 ,	0 ≤ y ≤ 2
1 ,	y > 2

(c)

Graph

f(y)

and

F(y).

(d)

Use

F(y)

to find

P(1 ≤ Y ≤ 2).

P(1 ≤ Y ≤ 2) =

(e)

Use

f(y)

and geometry to find

P(1 ≤ Y ≤ 2).

P(1 ≤ Y ≤ 2) =

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5. 1/10 points | Previous Answers WackerlyStat7 5.3.031. My Notes

The joint density function of

Y₁

and

Y₂

is given by

f(y₁, y₂) =

30y₁y₂²,	y₁ − 1 ≤ y₂ ≤ 1 − y₁, 0 ≤ y₁ ≤ 1,
0,	elsewhere.

(a)

Show that the marginal density of

Y₁

is a beta density with α = 2 and β = 4.

How would you derive the marginal density of

Y₁?

y₁ − 1

1 − y₁

(30y₁y₂²) dy₂

1 − y₁

y₁ − 1

(30y₁y₂²) dy₂

1

0

(30y₁y₂²) dy₁

1 − y₁

y₁ − 1

(30y₁y₂²) dy₁

1

0

(30y₁y₂²) dy₂

Give the marginal density of

Y₁.

f₁(y₁) =

where ≤ y₁ ≤

(b)

Derive the marginal density of

Y₂.

f₂(y₂) =

,	−1 ≤ y ≤ 0,
,	0 ≤ y ≤ 1

(c)

Derive the conditional density of

Y₂

given

Y₁ = y₁.

f(y₂ | y₁) =

, where y₁ + ≤ y₂ ≤ −y₁ +

(d)

Find

P(Y₂ > 0 | Y₁ = 0.42).

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6. 0/3 points | Previous Answers WackerlyStat7 7.2.015. My Notes

Suppose that

X₁, X₂,

, X_m and Y₁, Y₂,

, Y_n

are independent random samples, with the variables

X_i

normally distributed with mean

μ₁

and variance

σ₁²

and the variables

Y_i

normally distributed with mean

μ₂

and variance

σ₂².

The difference between the sample means,

X − Y,

is then a linear combination of

m + n

normally distributed random variables and, by this theorem, is itself normally distributed.

(a)

Find

E(X − Y).

(b)

Find

V(X − Y).

(c)

Suppose that

σ₁² = 6, σ₂² = 4.5, and m = n.

Find the minimum sample sizes so that

(X − Y)

will be within 1 unit of

(μ₁ − μ₂)

with probability 0.95. (Round your answer up to the nearest whole number.)

m = n =

You may need to use the appropriate appendix table or technology to answer this question.

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7. 0/4 points | Previous Answers WackerlyStat7 8.6.056. My Notes

Is America's romance with movies on the wane? In a poll of n = 800 randomly chosen adults, 45% indicated that movies were getting better whereas 42% indicated that movies were getting worse.

(a)

Find a 98% confidence interval for p, the overall proportion of adults who say that movies are getting better. (Round your answers to two decimal places.)

(b)

Does the interval include the value p = 0.50?

Yes No

Do you think that a majority of adults say that movies are getting better?

Yes. Because the value 0.50 is not contained in the interval, there not compelling evidence that a majority of adults feel movies are getting better. Yes. Because the value 0.50 is contained in the interval, there is compelling evidence that a majority of adults feel movies are getting better. No. Because the value 0.50 is contained in the interval, there is not compelling evidence that a majority of adults feel movies are getting better. No. Because the value 0.50 is not contained in the interval, there is not compelling evidence that a majority of adults feel movies are getting better.

You may need to use the appropriate appendix table or technology to answer this question.

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8. –/4 points WackerlyStat7 9.3.015. My Notes

Let

Y₁, Y₂,

, Y_n

denote a random sample from the uniform distribution on the interval (θ, θ + 1). Let

₁ = Y −

and

₂ = Y_(n) −

n + 1

Previously, we have confirmed that theta hat

₁ and

₂ are unbiased estimators of θ. Show that theta hat

₁ is a consistent estimator for θ.

lim n → ∞

₁)

lim n → ∞

Show that

₂ is a consistent estimator for θ.

lim n → ∞

_n)

lim n → ∞

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9. –/4 points WackerlyStat7 11.10.068.R. My Notes

The data in the following table follows the quadratic model

Y = β₀ + β₁x + β₂x² + ε.

y	x
1	−4
0	−3
0	−2
−1	−1
−1	0
0	1
0	2

(a)

Use R to fit the quadratic model

Y = β₀ + β₁x + β₂x² + ε

to the given data. (Round your numerical values to four decimal places.)

ŷ =

(b)

Use R to plot the fitted parabola.

The x y-coordinate plane is given. A scatter plot with a fitted curve has 7 points.
A pattern goes down and right becoming less steep from (−4, 1) to (−0.5, −1), then goes up and right becoming more steep from (−0.5, −1) to (2, 0).
The points are scattered moderately from the pattern.
The curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at approximately x = −2.79, changes direction at the point (−0.5, −0.75), goes up and right becoming more steep, crosses the y-axis at approximately y = −0.71, crosses the x-axis at approximately x = 1.79, and exits the window in the first quadrant.

The x y-coordinate plane is given. A scatter plot with a fitted line has 7 points.
A pattern goes down and right becoming less steep from (−4, 1) to (−0.5, −1), then goes up and right becoming more steep from (−0.5, −1) to (2, 0).
The points are scattered moderately from the pattern.
The line enters the window in the second quadrant, goes down and right, crosses the x-axis at x = −2, crosses the y-axis at approximately y = −0.29, and exits the window in the fourth quadrant.

(c)

Calculate SSE for the fitted parabola. (Round your answer to four decimal places.)

Calculate

S²

for the fitted parabola. (Round your answer to four decimal places.)

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10. –/8 points WackerlyStat7 12.3.010.R. My Notes

Two computers are compared by running a variety of benchmark programs and recording the difference in CPU time required to complete each program.

Six benchmark programs, run on two computers produced the following table of CPU times (in minutes). (Let population 1 be Computer 1 and population 2 be Computer 2. Use Computer 1 − Computer 2.)

Benchmarking Program	Computer 1	Computer 2
1.00	1.13	1.15
2.00	1.73	1.72
3.00	1.05	1.10
4.00	1.86	1.87
5.00	1.47	1.46
6.00	2.10	2.13

(a)

Do the data provided sufficient evidence to indicate a difference in mean CPU times required for the two computers to complete a job? Use R to test using α = 0.05.

State the null and alternative hypotheses (in minutes). (Enter != for ≠ as needed.)

H₀:

H_a:

State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to two decimal places.)

t > t <

Calculate the test statistic. (Round your answer to two decimal places.)

t =

What is the conclusion of your test?

Fail to reject H₀. There is insufficient evidence to indicate a difference in mean CPU times required for the two computers to complete a job. Fail to reject H₀. There is sufficient evidence to indicate a difference in mean CPU times required for the two computers to complete a job. Reject H₀. There is insufficient evidence to indicate a difference in mean CPU times required for the two computers to complete a job. Reject H₀. There is sufficient evidence to indicate a difference in mean CPU times required for the two computers to complete a job.

(b)

Use R to calculate the attained significance for this test. (Round your answer to four decimal places.)

(c)

Use R to calculate a 95% confidence interval for the difference in mean CPU time required for the two computers to complete a job. (Enter your answer using interval notation. Round your numerical values to four decimal places.)

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11. –/10 points WackerlyStat7 13.9.044.R. My Notes

Do average automobile insurance costs differ for different insurance companies? Other variables that impact insurance costs are geographic location, ages of the drivers, and type of coverage. Suppose the following are estimates (in dollars) of the cost of 6-month policies for basic liability coverage for a single man who has been licensed for 6–8 years, has no violations or accidents, and drives between 12,600 and 15,000 miles per year.

	Insurance Company
Location	A	B	C	D	E
1	734	745	668	1,075	1,202
2	834	725	618	859	1,172
3	1,490	1,384	1,214	1,492	1,682
4	998	884	802	1,581	1,272

(a)

Use R to determine if there is sufficient evidence to indicate that average insurance premiums differ from company to company. Use α = 0.05.

State the null and alternative hypotheses.

H₀: At least two of the average insurance premiums for the companies are the same.
H_a: All of the average insurance premiums for the companies differ.

H₀: β_j = 0

for all

j = 1, 2, 3, 4, 5

H_a: All of the average insurance premiums for the companies differ. H₀: At least one of the average insurance premiums for the companies are different.

H_a: β_j = 0

for all

j = 1, 2, 3, 4, 5

H₀: All of the average insurance premiums for the companies differ.
H_a: At least two of the average insurance premiums for the companies are the same.

H₀: β_j = 0

for all

j = 1, 2, 3, 4, 5

H_a: At least one of the average insurance premiums for the companies is different.

Calculate the test statistic. (Round your answer to two decimal places.)

F =

State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to two decimal places.)

F > F <

What is the conclusion of your test?

Fail to reject

H₀.

Thus, we find the data does not provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the five companies. Fail to reject

H₀.

Thus, we find the data does provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the five companies. Reject

H₀.

Thus, we find the data does provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the five companies. Reject

H₀.

Thus, we find the data does not provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the five companies.

(b)

Use R to determine if there is sufficient evidence to indicate that average insurance premiums differ from location to location. Use α = 0.05.

State the null and alternative hypotheses.

H₀: At least one of the average insurance premiums for the locations are different.

H_a: τ_i = 0

for all

i = 1, 2, 3, 4

H₀: τ_i = 0

for all

i = 1, 2, 3, 4

H_a: All of the average insurance premiums for the locations differ.

H₀: τ_i = 0

for all

i = 1, 2, 3, 4

H_a: At least one of the average insurance premiums for the locations is different. H₀: All of the average insurance premiums for the locations differ.
H_a: At least two of the average insurance premiums for the locations are the same. H₀: At least two of the average insurance premiums for the locations are the same.
H_a: All of the average insurance premiums for the locations differ.

Calculate the test statistic. (Round your answer to two decimal places.)

F =

State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to two decimal places.)

F > F <

What is the conclusion of your test?

Fail to reject

H₀.

Thus, we find the data does provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the four locations. Fail to reject

H₀.

Thus, we find the data does not provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the four locations. Reject

H₀.

Thus, we find the data does not provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the four locations. Reject

H₀.

Thus, we find the data does provide sufficient evidence to indicate that a real difference exists in the average insurance premiums for the four locations.

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