Deck 13: Inference About Comparing Two Populations

Question 1

The pooled proportion estimate is used when the null hypothesis states that the two population proportions differ by ____________________.

Answer 1

0
zero

Question 2

The difference in two sample proportions is a(n) ____________________ estimator of the difference in their respective population proportions.

Answer 2

unbiased
consistent

Question 3

In constructing a confidence interval estimate for the difference between two population proportions, we pool the population proportions when the populations are normally distributed.

Answer 3

False

Question 4

The pooled proportion estimate is used when the null hypothesis states that the two population proportions differ by some non-zero number.

Question 5

In constructing a confidence interval estimate for the difference between two population proportions, we ____________________ (always/sometimes/never) pool the population proportions.

Question 6

The difference in two sample proportions is an unbiased consistent estimator of the difference in their respective population proportions.

Question 7

The pooled proportion estimate is used when the proportion of successes from sample 1 equals the proportion of successes from sample 2.

Question 8

For testing the difference between two population proportions, the pooled proportion estimate should be used to compute the value of the test statistic when the:

A)populations are normally distributed.
B)sample sizes are small.
C)null hypothesis states that the two population proportions are equal.
D)samples are independently drawn from the populations.

Question 9

The variance of the difference in sample proportions equals the difference of their population variances.

Question 10

Pooling is made possible by hypothesizing (under the null hypothesis) that p₁ = p₂.

Question 11

The expected value of the difference between two sample proportions is the difference between their corresponding population proportions.

Question 12

The variance of the difference between two sample proportions equals the ____________________ of their population proportion variances.

Question 13

Which of the following is a required condition for using the normal approximation to the binomial in testing the difference between two population proportions?

A) $n _ { 1 } p _ { 1 } \geq 30 \text { and } n _ { 2 } p _ { 2 } \geq 30$
B) $n _ { 1 } p _ { 1 } \geq 5$ , $n _ { 1 } \left( 1 - p _ { 1 } \right) \geq 5$ , $n _ { 2 } p _ { 2 } \geq 5$ , and $n _ { 2 } \left( 1 - p _ { 2 } \right) \geq 5$
C) $n \tilde { p } _ { 1 } \geq 5$ , $n _ { 1 } \left( 1 - \tilde { p } _ { 1 } \right) \geq 5$ , $n _ { 2 } \tilde { p } _ { 2 } \geq 5$ , and $n _ { 2 } \left( 1 - \hat { p } _ { 2 } \right) \geq 5$
D)Choice b is the true requirement, but choice c is the one you actually check.

Question 14

The pooled proportion estimate is used when:

A)the proportion of successes from sample 1 equals the proportion of successes from sample 2.
B)the total number of successes in both samples divided by the total of both sample sizes equals 1.
C)the null hypothesis states that the two population proportions differ by some non-zero number.
D)None of these choices.

Question 15

Pooling is made possible by hypothesizing (under the null hypothesis) that p₁ __________ p₂.

Question 16

The pooled proportion estimate is found by taking the proportion of successes from sample 1 plus the proportion of successes from sample 2.

Question 17

A required condition for using the normal approximation to the binomial in testing the difference between two population proportions is that n₁p₁ $\ge$ 30 and n₂p₂ $\ge$ 30.

Question 18

In constructing a confidence interval estimate for the difference between two population proportions, we:

A)pool the population proportions when the populations are normally distributed.
B)pool the population proportions when the population means are equal.
C)pool the population proportions when they are equal.
D)never pool the population proportions.

Question 19

For testing the difference between two population proportions, the pooled proportion estimate is found by taking:

A)the proportion of successes from sample 1 plus the proportion of successes from sample 2.
B)the total number of successes in both samples divided by the total of both sample sizes.
C)the difference between the proportion of successes in each sample.
D)None of these choices.

Question 20

In comparing two population means the statistic under consideration is $\hat { p } _ { 1 } - \hat { p } _ { 2 }$ .

Question 21

NARRBEGIN: Worker Safety
Worker Safety
An OSHA agent wanted to determine if efforts to promote safety have been successful. By checking the records of 250 workers, he found that 30 of them suffered either minor or major injuries that year. A random sample of 400 workers last year revealed that 80 suffered some form of injury.NARREND

-{Worker Safety Narrative} What is the p-value of the test? Explain how to use it for testing the hypotheses.

Question 22

NARRBEGIN: Headache Medicine
Headache Medicine
A researcher wants to see if/how men and women differ in their reaction to a headache medicine with respect to drowsiness. In testing the hypotheses $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0.10$ vs. $H _ { 1 } : p _ { 1 } - p _ { 2 } < 0.10$ , the following statistics were obtained: n₁ = 400, x₁ = 208, n₂ = 250, and x₂ = 115, where x₁ and x₂ represent the number of patients in the two samples (men vs. women) who reported to have drowsiness as a result of taking headache medicine.NARREND

-{Headache Medicine Narrative} Estimate with 90% confidence the difference between the two population proportions.

Question 23

When the data from two populations are ____________________ the parameter to be tested and estimated is the difference between the two population proportions.

Question 24

NARRBEGIN: Worker Safety
Worker Safety
An OSHA agent wanted to determine if efforts to promote safety have been successful. By checking the records of 250 workers, he found that 30 of them suffered either minor or major injuries that year. A random sample of 400 workers last year revealed that 80 suffered some form of injury.NARREND

-{Worker Safety Narrative} Can the statistician infer at the 5% significance level that efforts to promote safety have been successful?

Question 25

If the sample sizes are large enough so the conditions are met, the difference between two sample proportions has an approximate ____________________ distribution.

Question 26

NARRBEGIN: TV Sex
TV Sex
A survey of 1,500 Canadians reveals that 945 believe that there is too much sex on television. In a survey of 1,500 Americans, 810 believe that there is too much television sex.NARREND

-{TV Sex Narrative} Estimate with 99% confidence the difference in the proportion of Canadians and Americans who believe that there is too much sex on television.

Question 27

NARRBEGIN: Senatorial Election
Senatorial Election
A political poll immediately prior to a senatorial election reveals that 145 out of 250 male voters and 105 out of 200 female voters intend to vote for the Democrat candidate.NARREND

-{Senatorial Election Narrative} Explain how to use the interval estimate to test the hypotheses.

Question 28

NARRBEGIN: Worker Safety
Worker Safety
An OSHA agent wanted to determine if efforts to promote safety have been successful. By checking the records of 250 workers, he found that 30 of them suffered either minor or major injuries that year. A random sample of 400 workers last year revealed that 80 suffered some form of injury.NARREND

-{Worker Safety Narrative} Estimate with 95% confidence the difference in population proportions.

Question 29

NARRBEGIN: Mass Production Line
Mass Production Line
A quality control examiner keeps a tally sheet of the number of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. $$\begin{array}{l}\begin{array} { |c | c c| } &\text { Froducts }\\\hline \text { Mass Production Line } & \text { Acceptable } & \text { Unacceptable } \\\hline 1 & 152 & 48 \\2 & 136 & 54 \\\hline\end{array}\end{array}$$ NARREND

-{Mass Production Line Narrative} Estimate with 95% confidence the difference in population proportions.

Question 30

NARRBEGIN: Senatorial Election
Senatorial Election
A political poll immediately prior to a senatorial election reveals that 145 out of 250 male voters and 105 out of 200 female voters intend to vote for the Democrat candidate.NARREND
{Senatorial Election Narrative} Can we infer at the 5% significance level that the proportions of male and female voters who intend to vote for the Democrat candidate differ?

Question 31

NARRBEGIN: TV Sex
TV Sex
A survey of 1,500 Canadians reveals that 945 believe that there is too much sex on television. In a survey of 1,500 Americans, 810 believe that there is too much television sex.NARREND
A councilwoman regularly polls her constituency to gauge her level of support among voters. This month, 652 out of 1158 voters support her. Five months ago, 412 out of 982 voters supported her. With a 5% significance level, can she infer that support has increased by at least 10 percentage points?

Question 32

NARRBEGIN: Mass Production Line
Mass Production Line
A quality control examiner keeps a tally sheet of the number of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. $$\begin{array}{l}\begin{array} { |c | c c| } &\text { Froducts }\\\hline \text { Mass Production Line } & \text { Acceptable } & \text { Unacceptable } \\\hline 1 & 152 & 48 \\2 & 136 & 54 \\\hline\end{array}\end{array}$$ NARREND

-{Mass Production Line Narrative} What is the p-value of the test? Explain how to use it for testing the hypotheses.

Question 33

NARRBEGIN: Headache Medicine
Headache Medicine
A researcher wants to see if/how men and women differ in their reaction to a headache medicine with respect to drowsiness. In testing the hypotheses $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0.10$ vs. $H _ { 1 } : p _ { 1 } - p _ { 2 } < 0.10$ , the following statistics were obtained: n₁ = 400, x₁ = 208, n₂ = 250, and x₂ = 115, where x₁ and x₂ represent the number of patients in the two samples (men vs. women) who reported to have drowsiness as a result of taking headache medicine.NARREND

-{Headache Medicine Narrative} What conclusion can we draw at the 10% significance level?

Question 34

NARRBEGIN: Senatorial Election
Senatorial Election
A political poll immediately prior to a senatorial election reveals that 145 out of 250 male voters and 105 out of 200 female voters intend to vote for the Democrat candidate.NARREND

-{Senatorial Election Narrative} Estimate with 95% confidence the difference in the proportion of male and female voters who intend to vote for the Democrat candidate.

Question 35

NARRBEGIN: Mass Production Line
Mass Production Line
A quality control examiner keeps a tally sheet of the number of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. NARREND
{Mass Production Line Narrative} Can the inspector infer at the 5% significance level that production line 1 is doing a better job than production line 2?

Question 36

NARRBEGIN: Speed Limits Violation
Speed Limits Violation
Do out-of-state motorists violate the speed limit more frequently than in-state motorists? This vital question was addressed by the highway patrol in a large eastern state. A random sample of the speeds of 2,500 randomly selected cars was categorized according to whether the car was registered in the state or in some other state and whether or not the car was violating the speed limit. The data follow. $$\begin{array} { | l | c c | } \hline & \text { In State Cars } & \text { Out af State Cars } \\\hline \text { Speeding } & 521 & 328 \\\text { Nat speeding } & 1141 & 510 \\\hline\end{array}$$ NARREND

-{Speed Limits Violation Narrative} Do these data provide enough evidence to support the highway patrol's claim at the 5% significance level?

Question 37

The expected value of the difference between two sample proportions is the ____________________ of/between their corresponding population proportions.

Question 38

NARRBEGIN: Senatorial Election
Senatorial Election
A political poll immediately prior to a senatorial election reveals that 145 out of 250 male voters and 105 out of 200 female voters intend to vote for the Democrat candidate.NARREND
{Senatorial Election Narrative} What is the p-value of the test?

Question 39

NARRBEGIN: TV Sex
TV Sex
A survey of 1,500 Canadians reveals that 945 believe that there is too much sex on television. In a survey of 1,500 Americans, 810 believe that there is too much television sex.NARREND
{TV Sex Narrative} Briefly explain what the interval estimate tells you.

Question 40

NARRBEGIN: TV Sex
TV Sex
A survey of 1,500 Canadians reveals that 945 believe that there is too much sex on television. In a survey of 1,500 Americans, 810 believe that there is too much television sex.NARREND
{TV Sex Narrative} Can we infer at the 99% significance level that the proportion of Canadians and Americans who believe that there is too much sex on television differ?

Question 41

The test statistic employed to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ is $F = s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ is F-distributed with v₁ = n₁ - 1 and v₂ = n₂ -1 degrees of freedom if the two populations are F-distributed.

Question 42

The statistical distribution used for testing the difference between two population variances is the

A)Student t-distribution
B)standard normal distribution
C)F-distribution
D)None of these choices.

Question 43

In testing for the equality of two population variances, when the populations are normally distributed, the 5% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of 0.05.

Question 44

The F-distribution is the sampling distribution of the ratio of:

A)two sample variances.
B)two normal population means.
C)two normal population variances.
D)None of these choices.

Question 45

NARRBEGIN: Speed Limits Violation
Speed Limits Violation
Do out-of-state motorists violate the speed limit more frequently than in-state motorists? This vital question was addressed by the highway patrol in a large eastern state. A random sample of the speeds of 2,500 randomly selected cars was categorized according to whether the car was registered in the state or in some other state and whether or not the car was violating the speed limit. The data follow. $$\begin{array} { | l | c c | } \hline & \text { In State Cars } & \text { Out af State Cars } \\\hline \text { Speeding } & 521 & 328 \\\text { Nat speeding } & 1141 & 510 \\\hline\end{array}$$ NARREND

-{Speed Limits Violation Narrative} Estimate with 95% confidence the difference in population proportions.

Question 46

When comparing two population variances, we use the ratio $\sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 }$ rather than the difference $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ .

Question 47

We use a t-test to determine whether two population variances are equal.

Question 48

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of:

A)0.90
B)0.20
C)0.10
D)0.05

Question 49

The ratio of two independent chi-squared variables divided by their degrees of freedom is:

A)normally distributed
B)Student t-distributed
C)chi-squared distributed
D)F-distributed

Question 50

The F-test used for testing the difference in 2 population variances is always a one-tailed test.

Question 51

All F-tests for the equality of two population variances are one-tailed tests.

Question 52

The F-distribution can only have non-negative values.

Question 53

Which of the following statements is false for an F-distribution?

A)Variables that are F-distributed range from 0 to $\infty$ .
B)The exact shape of the distribution is determined by two numbers of degrees of freedom.
C)The degrees of freedom for the numerator can be larger than, smaller than, or equal to the degrees of freedom for the denominator.
D)All of these choices are true.

Question 54

When testing the equality of two population variances the number in the null hypothesis is 0.

Question 55

The F-distribution is symmetric.

Question 56

NARRBEGIN: Speed Limits Violation
Speed Limits Violation
Do out-of-state motorists violate the speed limit more frequently than in-state motorists? This vital question was addressed by the highway patrol in a large eastern state. A random sample of the speeds of 2,500 randomly selected cars was categorized according to whether the car was registered in the state or in some other state and whether or not the car was violating the speed limit. The data follow. NARREND
{Speed Limits Violation Narrative} Briefly describe what the interval estimate tells you.

Question 57

The test for the equality of two population variances assumes that each of the two populations is normally distributed.

Question 58

When testing for the difference between two population variances with sample sizes of n₁ = 8 and n₂ = 10, the degrees of freedom are:

A)8 and 10
B)7 and 9
C)2 and 18
D)18 and 2

Question 59

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.05 to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ . The two sample variances are $s _ { 1 } ^ { 2 } = 400$ and $s _ { 2 } ^ { 2 } = 400$ , and the sample sizes are n₁ = 25 and n₂ = 25. The calculated value of the test statistic will be F = 2.

Question 60

Which of the following statements is correct regarding the percentile points of the F-distribution?

A)F_0.05,10,20 = 1/F_0.95,10,20
B)F_0.05,10,20 = 1/F_0.05,20,10
C)F_0.95,10,20 = 1/F_0.05,20,10
D)F_0.95,10,20 = 1/F_0.95,20,10

Question 61

The sampling distribution of the ratio of two sample variances is said to be F-distributed provided that we have two ____________________ samples drawn from their respective populations.

Question 62

When testing for the equality of two population variances the number in the null hypothesis is ____________________.

Question 63

NARRBEGIN: Clinic Waiting Time
Clinic Waiting Time
In a random sample of 20 patients who visited a clinic at Medical Center 1, a researcher found that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the clinic of Medical Center 2, the researcher found the variance to be 178.8.NARREND

-{Clinic Waiting Time Narrative} Can we infer at the 5% level of significance that the population variances differ?

Question 64

NARRBEGIN: Profit Margin
Profit Margin
An investor is considering two types of investment. She is quite satisfied that the expected profit margin on Investment 1 is higher than the expected profit margin on Investment 2. However, she is quite concerned that the risk associated with Investment 1 is higher than that of Investment 2. To help make her decision, she randomly selects seven monthly profit margins on investment 1 and ten monthly profit margins on investment 2. She finds that the sample variances of Investments 1 and 2 are 225 and 118, respectively.NARREND
{Profit Margin Narrative} Estimate with 95% confidence the ratio of the two population variances.

Question 65

The test statistic for testing for the equality of two population variances has a(n) ____________________ distribution.

Question 66

NARRBEGIN: Profit Margin
Profit Margin
An investor is considering two types of investment. She is quite satisfied that the expected profit margin on Investment 1 is higher than the expected profit margin on Investment 2. However, she is quite concerned that the risk associated with Investment 1 is higher than that of Investment 2. To help make her decision, she randomly selects seven monthly profit margins on investment 1 and ten monthly profit margins on investment 2. She finds that the sample variances of Investments 1 and 2 are 225 and 118, respectively.NARREND
{Profit Margin Narrative} Briefly describe what the interval estimate tells you.

Question 67

To estimate the ratio of the population variances you use the ____________________ of the ____________________ variances.

Question 68

NARRBEGIN: Profit Margin
Profit Margin
An investor is considering two types of investment. She is quite satisfied that the expected profit margin on Investment 1 is higher than the expected profit margin on Investment 2. However, she is quite concerned that the risk associated with Investment 1 is higher than that of Investment 2. To help make her decision, she randomly selects seven monthly profit margins on investment 1 and ten monthly profit margins on investment 2. She finds that the sample variances of Investments 1 and 2 are 225 and 118, respectively.NARREND
{Profit Margin Narrative} Can she infer at the 5% significance level that the population variance of investment 1 exceeds that of investment 2?

Question 69

The test statistic for testing for the equality of two population variances has an F-distribution with ____________________ and ____________________ degrees of freedom.

Question 70

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is:

A)normal
B)Student-t
C)F
D)chi-squared

Question 71

We compare two population variances by examining their ____________________.

Question 72

Random samples from two normal populations produced the following statistics: , , , and . Is there enough evidence at the 5% significance level to infer that the variance of Population 1 is larger than the variance of Population 2?

Question 73

NARRBEGIN: Fitness Program
Fitness Program
A statistician wants to test for the equality of means in two independent samples drawn from normal populations of people enrolled in a fitness program. However, he will not perform the equal-variance t-test of the difference between the population means if the condition necessary for its use is not satisfied. The number of pound lost at the completion of the program data follow:
$$\begin{array} { | l | c c c c c c c c c c | } \hline \text { Sample 1 } & 7 & 9 & 6 & 15 & 7 & 10 & 8 & 12 & \\\hline \text { Sample } 2 & 2 & 25 & 9 & 15 & 10 & 18 & 5 & 22 & 27 & 3 \\\hline\end{array}$$ NARREND

-{Fitness Program Narrative} Estimate with 95% confidence the ratio of the two population variances and interpret.

Question 74

The sampling distribution of the ratio of two (independent) sample variances is said to be ____________________ distributed.

Question 75

If the F-test statistic is large, that means the variance of Population 1 is ____________________ than/to the variance of Population 2.

Question 76

NARRBEGIN: Clinic Waiting Time
Clinic Waiting Time
In a random sample of 20 patients who visited a clinic at Medical Center 1, a researcher found that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the clinic of Medical Center 2, the researcher found the variance to be 178.8.NARREND
{Clinic Waiting Time Narrative} Estimate with 95% confidence the ratio of the two population variances and interpret.

Question 77

The test for the equality of two population variances is based on the:

A)difference between the two sample variances.
B)ratio of the two sample variances.
C)sum of the two sample variances.
D)product of the two sample variances.

Question 78

NARRBEGIN: Anioxidants
Antioxidants
A food processor wants to compare two antioxidants for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with antioxidant A and 16 are treated with antioxidant B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below
NARREND
{Antioxidants Narrative} State the null and alternative hypotheses for determining if the population variances differ for Antioxidants A and B.

Question 79

NARRBEGIN: Fitness Program
Fitness Program
A statistician wants to test for the equality of means in two independent samples drawn from normal populations of people enrolled in a fitness program. However, he will not perform the equal-variance t-test of the difference between the population means if the condition necessary for its use is not satisfied. The number of pound lost at the completion of the program data follow:
NARREND
{Fitness Program Narrative} Can the statistician conclude at the 5% significance level that the required condition is not satisfied?

Question 80

The sampling distribution of the ratio of two sample variances $s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ is said to be F-distributed provided that:

A)the samples are independent from any distributions.
B)the populations are normal with equal variances.
C)the samples are matched and their sizes are large.
D)the samples are independently drawn from two normal populations.