Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of39arrests last month,24were of males aged 15 to 34 years. Use a1%level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀:p< 0 .7;H₁:p= 0.7

H₀:p0.7;H₁:p= 0.7

H₀:p= 0.7;H₁:p> 0.7

H₀:p= 0 .7;H₁:p< 0.7

H₀:p= 0.7;H₁:p0.7

(b) What sampling distribution will you use?

The standard normal, sincenp< 5 andnq< 5.

The Student'st, sincenp< 5 andnq< 5.

The standard normal, sincenp> 5 andnq> 5.

The Student'st, sincenp> 5 andnq> 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(c) Find theP-value of the test statistic. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to theP-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

At the= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

There is insufficient evidence at the 0.01 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

8.

[-/4.08 Points]

DETAILS

BBUNDERSTAT12 8.3.010.MI.S.

MY NOTES

ASK YOUR TEACHER

Women athletes at a certain university have a long-term graduation rate of 67%. Over the past several years, a random sample of40women athletes at the school showed that23eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the university is now less than 67%? Use a1% level of significance.

(a)

What is the level of significance?

(b)

State the null and alternate hypotheses.

H₀:p= 0.67;H₁:p> 0.67

H₀:p< 0.67;H₁:p= 0.67

H₀:p= 0.67;H₁:p0.67

H₀:p= 0.67;H₁:p< 0.67

What sampling distribution will you use?

The standard normal, sincenp> 5 andnq> 5.

The Student'st, sincenp> 5 andnq> 5.

The standard normal, sincenp< 5 andnq< 5.

The Student'st, sincenp< 5 andnq< 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(c)

Find theP-value of the test statistic. (Round your answer to four decimal places.)

(d)

Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

At the= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e)

Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the true proportion of women athletes who graduate is less than 0.67.

There is insufficient evidence at the 0.01 level to conclude that the true proportion of women athletes who graduate is less than 0.67.

9.

[-/4.08 Points]

DETAILS

BBUNDERSTAT12 8.4.009.MI.S.

MY NOTES

ASK YOUR TEACHER

In this problem, assume that the distribution of differences is approximately normal.Note: For degrees of freedomd.f. not in the Student'sttable, use the closestd.f. that issmaller. In some situations, this choice ofd.f. may increase theP-value by a small amount and therefore produce a slightly more "conservative" answer.

Are America's top chief executive officers (CEOs) really worth all that money? One way to answer this question is to look at rowB, the annual company percentage increase in revenue, versus rowA, the CEO's annual percentage salary increase in that same company. Suppose a random sample of companies yielded the following data:

B: Percent increase for company

24 25 27 18 6 4 21 37

A: Percent increase for CEO

23 25 22 14 4 19 15 30

Do these data indicate that the population mean percentage increase in corporate revenue (rowB) is different from the population mean percentage increase in CEO salary? Use a 5% level of significance. (Letd=BA.)

(a)

What is the level of significance?

State the null and alternate hypotheses.

H₀:_d= 0;H₁:_d0

H₀:_d= 0;H₁:_d> 0

H₀:_d0;H₁:_d= 0

H₀:_d= 0;H₁:_d< 0

H₀:_d> 0;H₁:_d= 0

(b)

What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume thatdhas an approximately uniform distribution.

The Student'st. We assume thatdhas an approximately normal distribution.

The Student'st. We assume thatdhas an approximately uniform distribution.

The standard normal. We assume thatdhas an approximately normal distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)

(c)

Find (or estimate) theP-value.

P-value > 0.500

0.250 <P-value < 0.500

0.100 <P-value < 0.250

0.050 <P-value < 0.100

0.010 <P-value < 0.050

P-value < 0.010

(d)

Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

Since theP-value >, we fail to rejectH₀. The data are not statistically significant.

Since theP-value, we fail to rejectH₀. The data are statistically significant.

Since theP-value >, we rejectH₀. The data are not statistically significant.

Since theP-value, we rejectH₀. The data are statistically significant.

(e)

Interpret your conclusion in the context of the application.

RejectH₀. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

Fail to rejectH₀. At the 5% level of significance, the evidence is sufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

Fail to rejectH₀. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

RejectH₀. At the 5% level of significance, the evidence is sufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

10.

[-/4.08 Points]

DETAILS

BBUNDERSTAT12 8.4.010.S.

MY NOTES

ASK YOUR TEACHER

In this problem, assume that the distribution of differences is approximately normal.Note: For degrees of freedomd.f. not in the Student'sttable, use the closestd.f. that issmaller. In some situations, this choice ofd.f. may increase theP-value by a small amount and therefore produce a slightly more "conservative" answer.

Is fishing better from a boat or from the shore? Pyramid Lake is located on the Paiute Indian Reservation in Nevada. Presidents, movie stars, and people who just want to catch fish go to Pyramid Lake for really large cutthroat trout. Let rowBrepresent hours per fish caught fishing from the shore, and let rowArepresent hours per fish caught using a boat. The following data are paired by month from October through April.

Oct Nov Dec Jan Feb March April

B: Shore 1.5 1.8 2.0 3.2 3.9 3.6 3.3

A: Boat 1.4 1.4 1.7 2.2 3.3 3.0 3.8

Use a 1% level of significance to test if there is a difference in the population mean hours per fish caught using a boat compared with fishing from the shore. (Letd=BA.)

(a) What is the level of significance?

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀:_d= 0;H₁:_d0; two-tailed

H₀:_d= 0;H₁:_d< 0; left-tailed

H₀:_d0;H₁:_d= 0; two-tailed

H₀:_d= 0;H₁:_d> 0; right-tailed

(b) What sampling distribution will you use? What assumptions are you making?

The Student'st. We assume thatdhas an approximately uniform distribution.

The standard normal. We assume thatdhas an approximately normal distribution.

The standard normal. We assume thatdhas an approximately uniform distribution.

The Student'st. We assume thatdhas an approximately normal distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)

(c) Find (or estimate) theP-value.

P-value > 0.500

0.250 <P-value < 0.500

0.100 <P-value < 0.250

0.050 <P-value < 0.100

0.010 <P-value < 0.050

P-value < 0.010

Sketch the sampling distribution and show the area corresponding to theP-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

(e) State your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

Reject the null hypothesis, there is insufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

Fail to reject the null hypothesis, there is insufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

Reject the null hypothesis, there is sufficient evidence to claim a difference in population mean hours per fish between boat fishing and shore fishing.

11.

[-/4.08 Points]

DETAILS

BBUNDERSTAT12 8.5.013.MI.S.

For one binomial experiment,n₁= 75

binomial trials producedr₁=30

successes. For a second independent binomial experiment,n₂= 100

binomial trials producedr₂=50

successes.At the 5% level of significance, test the claim that the probabilities of success for the two binomial experiments differ.

(a)

Compute the pooled probability of success for the two experiments. (Round your answer to three decimal places.)

(b)

Check Requirements: What distribution does the sample test statistic follow? Explain.

The standard normal. The number of trials is sufficiently large.

The Student'st. We assume the population distributions are approximately normal.

The Student'st. The number of trials is sufficiently large.

The standard normal. We assume the population distributions are approximately normal.

(c)

State the hypotheses.

H₀:p₁=p₂;H₁:p₁<p₂

H₀:p₁<p₂;H₁:p₁=p₂

H₀:p₁=p₂;H₁:p₁p₂

H₀:p₁=p₂;H₁:p₁>p₂

(d)

Computep₁p₂.

p₁p₂=

Compute the corresponding sample distribution value. (Test the differencep₁p₂. Do not use rounded values. Round your final answer to two decimal places.)

(e)

Find theP-value of the sample test statistic. (Round your answer to four decimal places.)

(f)

Conclude the test.

At the= 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(g)

Interpret the results.

Reject the null hypothesis, there is insufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.

Reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.

Fail to reject the null hypothesis, there is insufficient evidence that the probabilities of success for the two binomial experiments differ.

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.

12.

[-/4.08 Points]

DETAILS

BBUNDERSTAT12 8.5.015.

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults. Assume that REM sleep time is normally distributed for both children and adults. A random sample ofn₁=8children (9 years old) showed that they had an average REM sleep time ofx₁=2.9hours per night. From previous studies, it is known that₁=0.6hour. Another random sample ofn₂=8adults showed that they had an average REM sleep time ofx₂=2.10hours per night. Previous studies show that₂=0.7hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀:₁=₂;H₁:₁₂

H₀:₁=₂;H₁:₁>₂

H₀:₁=₂;H₁:₁<₂

H₀:₁<₂;H₁:₁=₂

(b) What sampling distribution will you use? What assumptions are you making?

The Student'st. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The Student'st. We assume that both population distributions are approximately normal with known standard deviations.

What is the value of the sample test statistic? (Test the difference₁₂. Round your answer to two decimal places.)

(c) Find (or estimate) theP-value. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to theP-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

13.

[-/4.17 Points]

DETAILS

BBUNDERSTAT12 8.5.020.S.

A random sample ofn₁= 16 communities in western Kansas gave the following information for people under 25 years of age.

x₁:Rate of hay fever per 1000 population for people under 25

100 92 121 127 93 123 112 93

125 95 125 117 97 122 127 88

A random sample ofn₂= 14 regions in western Kansas gave the following information for people over 50 years old.

x₂:Rate of hay fever per 1000 population for people over 50

93 108 100 96 112 88 110

79 115 100 89 114 85 96

(i) Use a calculator to calculatex₁,s₁,x₂, ands₂. (Round your answers to four decimal places.)

x₁=

s₁=

x₂=

s₂=

(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use= 0.05.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀:₁=₂;H₁:₁<₂

H₀:₁>₂;H₁:₁=₂

H₀:₁=₂;H₁:₁₂

H₀:₁=₂;H₁:₁>₂

(b) What sampling distribution will you use? What assumptions are you making?

The Student'st. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The Student'st. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

What is the value of the sample test statistic? (Test the difference₁₂. Round your answer to three decimal places.)

(c) Find (or estimate) theP-value.

P-value > 0.250

0.125 <P-value < 0.250

0.050 <P-value < 0.125

0.025 <P-value < 0.050

0.005 <P-value < 0.025

P-value < 0.005

Sketch the sampling distribution and show the area corresponding to theP-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is sufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Reject the null hypothesis, there is insufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Fail to reject the null hypothesis, there is insufficient evidence that the mean rate of hay fever is lower for the age group over 50.

Fail to reject the null hypothesis, there is sufficient evidence that the mean rate of hay fever is lower for the age group over 50.