Homework answers / question archive / Sanford-Brown College - MATH 2014 CHAPTER 13 SECTION 1-2: INFERENCE ABOUT COMPARING TWO POPULATIONS 1)Independent samples are those for which the selection process for one is not related to the selection process for the other

Sanford-Brown College - MATH 2014 CHAPTER 13 SECTION 1-2: INFERENCE ABOUT COMPARING TWO POPULATIONS 1)Independent samples are those for which the selection process for one is not related to the selection process for the other

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Sanford-Brown College - MATH 2014

CHAPTER 13 SECTION 1-2: INFERENCE ABOUT COMPARING TWO POPULATIONS

1)Independent samples are those for which the selection process for one is not related to the selection process for the other.

 

 

 

 

 

     2.   In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference

 is normal if the sample sizes are both greater than 30.

 

 

 

 

 

     3.   The pooled-variances t-test requires that the two population variances need not be the same.

 

 

 

 

     4.   In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference

 if the populations are normal with equal variances.

 

 

 

 

 

     5.   A political analyst in Iowa surveys a random sample of registered Republicans and compares the results with those obtained from a random sample of registered Democrats . This would be an example of two independent samples.

 

 

 

 

     6.   Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 44.

 

 

 

 

     7.   The sampling distribution of

 is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n₁ and n₂ are large.

 

 

 

 

 

     8.   When the population variances are unequal, we estimate each population variance with its sample variance. Hence, the unequal-variances test statistic of

 is approximately Student t-distributed with n₁ + n₂ ? 2 degrees of freedom.

 

 

 

 

 

     9.   Unless we can conclude that the population variances are equal, we cannot use the pooled variance estimate.

 

 

 

 

   10.   The equal-variances test statistic of

 is Student t-distributed with n₁ + n₂ degrees of freedom, provided that the two populations are normal.

 

 

 

 

 

   11.   Both the equal-variances and unequal variances test statistic and confidence interval estimator of

 require that the two populations be normally distributed.

 

 

 

 

 

   12.   When the sample sizes are equal, the pooled variance of the two samples is the average of the two sample variances.

 

 

 

 

   13.   The expected value of

 is

.

 

 

 

 

 

   14.   The best estimator of the difference between two population means

 is the difference between two sample means

.

 

 

 

 

 

   15.   When we test for differences between the means of two independent populations, we can only use a two-tailed test.

 

 

 

 

   16.   The variance of

 is

.

 

 

 

 

 

MULTIPLE CHOICE

 

   17.   The expected value of the difference of two sample means equals the difference of the corresponding population means when:

a.	the populations are normally distributed.
b.	the samples are independent.
c.	the populations are approximately normal and the sample sizes are large.
d.	All of these choices are true.

 

 

 

 

 

   18.   In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you:

a.	round to the nearest integer.
b.	change the sample sizes until the number of degrees of freedom becomes an integer.
c.	assume that the population variances are equal, and then use df = n1 + n2 ? 2.
d.	None of these choices.

 

 

 

 

 

   19.   The quantity

 is called the pooled variance estimate of the common variance of two unknown but equal population variances. It is the weighted average of the two sample variances, where the weights represent the:

 

a.	sample variances.
b.	sample standard deviations.
c.	degrees of freedom for each sample.
d.	None of these choices.

 

 

 

 

 

   20.   Two independent samples of sizes 20 and 30 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. In order to test the difference between the population means,

, the sampling distribution of the sample mean difference,

, is:

 

a.	normal.
b.	Student-t with 50 degrees of freedom.
c.	Student-t with 48 degrees of freedom.
d.	None of these choices.

 

 

 

 

 

   21.   Two independent samples of sizes 40 and 50 are randomly selected from two populations to test the difference between the population means

. Assume the population variances are known. The sampling distribution of the sample mean difference

 is:

 

a.	normally distributed.
b.	approximately normal.
c.	Student t-distributed with 88 degrees of freedom.
d.	None of these choices.

 

 

 

 

 

   22.   Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with equal variances (assumed to be unknown). In order to test the difference between the population means, the test statistic is:

a.	a standard normal random variable.
b.	approximately standard normal random variable.
c.	Student t-distributed with 58 degrees of freedom.
d.	Student t-distributed with 33 degrees of freedom.

 

 

 

 

 

   23.   In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference

 if:

 

a.	the sample sizes are both large.
b.	the populations are normal with equal variances.
c.	the populations are non-normal with unequal variances.
d.	All of these choices are true.

 

 

 

 

 

   24.   In testing the difference between two population means for which the population variances are unknown and not assumed to be equal, two independent samples are drawn from the populations. Which of the following tests is appropriate?

a.	z-test
b.	pooled-variances t-test
c.	unequal variances t-test
d.	None of these choices.

 

 

 

 

 

   25.   In testing the difference between the means of two normal populations using two independent samples when the population variances are unequal, the sampling distribution of the resulting statistic is:

a.	normal.
b.	Student-t.
c.	approximately normal.
d.	approximately Student-t.

 

 

 

 

 

   26.   In constructing a confidence interval estimate for the difference between the means of two independent normally distributed populations, we:

a.	pool the sample variances when the unknown population variances are equal.
b.	pool the sample variances when the population variances are known and equal.
c.	pool the sample variances when the population means are equal.
d.	never pool the sample variances.

 

 

 

 

 

   27.   The t-test for the difference between the means of two independent populations assumes that the respective:

a.	sample sizes are equal.
b.	populations are normal.
c.	means are equal.
d.	All of these choices are true.

 

 

 

 

 

   28.   If we are testing for the difference between the means of two independent populations with equal variances, samples of n₁ = 15 and n₂ = 15 are taken, then the number of degrees of freedom is equal to

a.	13
b.	14
c.	28
d.	29

 

 

 

 

 

   29.   In testing for the differences between the means of two independent populations where the variances in each population are unknown but assumed equal, the degrees of freedom is:

a.	n1 + n2
b.	n1 + n2 ? 2
c.	n1 + n2 ? 1
d.	None of these choices

 

 

 

 

 

   30.   Given the information:

 the number of degrees of freedom that should be used in the pooled variance t-test is:

 

a.	40
b.	38
c.	15
d.	25

 

 

 

 

 

   31.   When testing

 vs.

, the observed value of the z-score was found to be ?2.15. Then, the p-value for this test would be

 

a.	.0158
b.	.0316
c.	.9842
d.	.9684

 

 

 

 

 

   32.   A political analyst in Hawaii surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of:

a.	independent samples.
b.	dependent samples.
c.	independent samples only if the sample sizes are equal.
d.	dependent samples only if the sample sizes are equal.

 

 

 

 

 

   33.   In testing for differences between the means of two independent populations the null hypothesis is:

a.
b.
c.
d.

 

 

 

 

 

   34.   Suppose we randomly selected 250 people, and on the basis of their responses to a survey we assigned them to one of two groups: high-risk group and low-risk group. We then recorded the blood pressure for the members of each group. Such data are called:

a.	observational.
b.	experimental.
c.	matched.
d.	None of these choices.

 

 

 

 

 

COMPLETION

 

   35.   When the sample sizes are equal, the pooled variance of the two samples is the ____________________ of the two sample variances.

 

 

          

 

 

   36.   The equal-variances test statistic of

 is Student t-distributed with n₁ + n₂ ? 2 degrees of freedom provided that the two populations are ____________________.

 

 

 

          

 

 

   37.   The unequal-variances test statistic of

 has an approximate ____________________ distribution with n₁ + n₂ ? 2 degrees of freedom.

 

 

 

          

 

 

   38.   ____________________ samples are those for which the selection process for one is not related to the selection process for the other.

 

 

          

 

 

   39.   When two population variances are ____________________ we estimate each population variance with its sample variance. The test statistic of

 is approximately Student t-distributed with n₁ + n₂ ? 2 degrees of freedom.

 

 

 

          

 

 

   40.   A political analyst in Iowa surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of ____________________ samples.

 

 

          

 

 

   41.   The pooled-variances t-test is used when the two population variances are ____________________.

 

 

          

 

 

   42.   When the population variances are unknown and unequal, we estimate each population variance with its ____________________ variance.

 

 

          

 

 

   43.   The pooled variance estimator is the ____________________ average of the two sample variances.

 

 

          

 

 

SHORT ANSWER

 

Aptitude Test Scores

 

Two random samples of 40 students were drawn independently from two populations of students. Assume their aptitude tests are normally distributed (total points = 100). The following statistics regarding their scores in an aptitude test were obtained:

.

 

 

 

   44.   {Aptitude Test Scores Narrative} Test at the 5% significance level to determine whether we can infer that the two population means differ.

 

 

          

 

 

   45.   {Aptitude Test  Scores Narrative} Estimate with 95% confidence the difference between the two population means.

 

 

          

 

 

   46.   {Aptitude Test  Scores Narrative} Explain how to use the 95% confidence interval to test the hypotheses at  = .05.

 

 

          

 

 

Starting Salary

 

In testing the hypotheses

 vs.

, two random samples from two populations of college of business graduates majoring in global marketing and international business produced the following statistics regarding their starting salaries (in $1000s):

,

,

,

,

, and

. (Assume the salaries have normal distributions.)

 

 

 

   47.   {Starting Salary Narrative} What conclusion can we draw at the 5% significance level?

 

 

          

 

 

   48.   {Starting Salary Narrative} Estimate with 95% confidence the difference between the two population means.

 

 

          

 

 

   49.   {Starting Salary Narrative} Explain how to use the 95% confidence interval to test the hypotheses at  = .05.

 

 

          

 

 

   50.   The service manager of a car dealer wants to determine if owners of new cars (two years old or less) tune up their cars more frequently than owners of older cars (more than two years old). From his records he takes a random sample of ten new cars and ten older cars and determines the number of times the cars were tuned up in the last 12 months. The data follow. Do these data allow the service station owner to infer at the 10% significance level that new car owners tune up their cars more frequently than older car owners?

 

Frequency of Tune Ups in Past 12 Months
New Car Owners	Old Cars Owners
6	4
3	2
3	1
3	2
4	3
3	2
6	2
5	3
5	2
4	1

 

 

          

 

 

Undergraduates’ Test Scores

 

35 undergraduate students who completed two years of college were asked to take a basic mathematics test. The mean and standard deviation of their scores were 75.1 and 12.8, respectively. In a random sample of 50 students who only completed high school, the mean and standard deviation of the test scores were 72.1 and 14.6, respectively.

 

 

   51.   {Undergraduates’ Test Scores Narrative} Can we infer at the 10% significance level that a difference exists between the two groups?

 

.

 

          

 

 

   52.   {Undergraduates’ Test Scores Narrative} Estimate with 90% confidence the difference in mean scores between the two groups of students.

 

 

 

          

 

 

   53.   {Undergraduates’ Test Scores Narrative} Explain how to use the interval estimate to test the hypotheses.

 

 

          

 

 

Additives

 

A food processor wants to compare two additives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with additive A and 16 are treated with additive 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

 

	Additive A	Additive B
Sample Mean	108.7 hours	98.7 hours
Sample Standard Deviation	10.5 hours	13.6 hours

 

 

 

   54.   {Additives Narrative} State the null and alternative hypotheses to determine if the average number of hours until spoilage begins differs for the additives A and B.

 

 

          

 

 

   55.   {Additives Narrative} Assume population variances are equal. Calculate the pooled variance and the value of the test statistic.

 

 

          

 

 

   56.   {Additives Narrative} Determine the rejection region at ? = .05 and write the proper conclusion.