Exam 9: Inferences From Two Samples

In a random sample of 500 people aged $20 - 24,22 \%$ were smokers. In a random sample of 450 people aged 25-29, 14\% were smokers. A $95 \%$ confidence interval for the difference between the proportion of 20-24 year olds and the proportion of 25-29 year olds who are smokers is $0.032 < \mathrm { p } _ { 1 } - \mathrm { p } _ { 2 } < 0.128$ Which of the following statements give a correct interpretation of this confidence interval? I. We can be $95 \%$ confident that the interval $0.032$ to $0.128$ contains the true difference between the two population proportions. II. There is a $95 \%$ chance that the true difference between the two population proportions lies between $0.032$ and $0.128$ . III. If the process were repeated many times, each time selecting random samples of 500 people aged 20-24 and 450 people aged 25-29 and each time constructing a confidence interval for $\mathrm { p } _ { 1 }$ $p _ { 2 } , 95 \%$ of the time the true difference between the two population proportions will lie between $0.032$ and $0.128$ . IV. If the process were repeated many times, each time selecting random samples of 500 people aged 20-24 and 450 people aged 25-29 and each time constructing a confidence interval for $p _ { 1 } -$ $\mathrm { p } _ { 2 } , 95 \%$ of the time the confidence interval limits will contain the true difference between the two population proportions.

Consider the set of differences between two dependent sets: 84, 85, 83, 63, 61, 100, 98. Round to the nearest tenth.

A hypothesis test is to be performed to test the equality of two population means. The sample sizes are large and the samples are independent. Give an expression for the population standard deviation of the $\left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right)$ values in terms of $\mathrm { s } _ { 1 } , \mathrm {~s} _ { 2 } , \mathrm { n } _ { 1 }$ , and $\mathrm { n } _ { 2 }$ .

The table shows the number satisfied in their work in a sample of working adults with a college education and in a sample of working adults without a college education. Assume that you plan to use a significance level of $\alpha = 0.05 \text { to test the claim that } \mathrm { p } _ { 1 } > \mathrm { p } _ { 2 }$ Find the critical value(s) for this hypothesis test. Do the data provide sufficient evidence that a greater proportion of those with a college education are satisfied in their work? College Education No College Education Number in sample 194 188 Number satisfied in their work 74 70

Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected - Use the given sample data to test the claim that p1 > p2. Use a significance level of 0.01 . =85 =90 =38 =23

Perform the indicated hypothesis test. Assume that the two samples are independent simple random samples selected from normally distributed populations. Also assume that the population standard deviations are equal (?1 = ?2), so that the standard error of the difference between means is obtained by pooling the sample variances . -A researcher was interested in comparing the resting pulse rates of people who exercise regularly and the pulse rates of those who do not exercise regularly. Independent simple random samples of 16 people who do not exercise regularly and 12 people who exercise regularly were selected, and the resting pulse rates (in beats per minute) were recorded. The summary statistics are as follows. Do Not Exercise Do Exercise =73.9 beats / =68.7 beat / =10.9 beats / =8.7 beats / =16 =12 Use a 0.025 significance level to test the claim that the mean resting pulse rate of people who do not exercise regularly is greater than the mean resting pulse rate of people who exercise regularly. Use the traditional method of hypothesis testing.

Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected. -In a random sample of 300 women, $47 \%$ favored stricter gun control legislation. In a random sample of 200 men, $29 \%$ favored stricter gun control legislation. Construct a $98 \%$ confidence interval for the difference between the population proportions $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$ .

Assume that you plan to use a significance level of $\alpha = 0.05 \text { to test the claim that } \mathrm { p } _ { 1 } = \mathrm { p } _ { 2 }$ , Use the given sample sizes and numbers of successes to find the P-value for the hypothesis test. - =50 =50 =8 =7

When testing for a difference between the means of a treatment group and a placebo group, the computer display below is obtained. Using a 0.05 significance level, is there sufficient evidence to support the claim that the treatment group (variable 1) comes from a population with a mean that is less than the mean for the placebo population? Explain. t-Test: Two Sample for Means 1 Variable 1 Variable 2 2 Mean 65.10738 66.18251 3 Known Variance 8.102938 10.27387 4 Observations 50 50 5 Hypothesized Mean Difference 0 6 -1.773417 7 P(T<=t) one-tail 0.0384 8 TCritical one-tail 1.644853 9 P(T<=t) two-tail 0.0768 10 tCritical two-tail 1.959961

Construct a confidence interval for µd, the mean of the differences d for the population of paired data. Assume that the population of paired differences is normally distributed. -A test of abstract reasoning is given to a random sample of students before and after they completed a formal logic course. The results are given below. Construct a 95% confidence interval for the mean difference between the before and after scores. Before 74 83 75 88 84 63 93 84 91 77 After 73 77 70 77 74 67 95 83 84 75

To test the null hypothesis that the difference between two population proportions is equal to a nonzero constant c, use the test statistic $z = \frac { \left( \hat {p} _ { 1 } - \hat {p} _ { 2 } \right) - c } { \sqrt { \hat { p } _ { 1 } ( 1 - \hat { p } 1 ) / n _ { 1 } + \hat { p } _ { 2 } \left( 1 - \hat { p _ { 2 } } \right) / \mathrm { n } _ { 2 } } }$ As long as n1 and n2 are both large, the sampling distribution of the test statistic z will be approximately the standard normal distribution. Given the sample data below, test the claim that the proportion of male voters who plan to vote Republican at the next presidential election is 15 percentage points more than the percentage of female voters who plan to vote Republican. Use the P-value method of hypothesis testing and use a significance level of 0.10. Men: $\mathrm { n } _ { 1 } = 250 , \mathrm { x } _ { 1 } = 146$ Women: $\mathrm { n } _ { 2 } = 202 , \mathrm { x } _ { 2 } = 103$

Construct a confidence interval for µd, the mean of the differences d for the population of paired data. Assume that the population of paired differences is normally distributed. -Using the sample paired data below, construct a $90 \%$ confidence interval for the population mean of all differences $\mathrm { x } - \mathrm { y }$ . 3.3 6.8 5.9 4.3 7.6 3.0 5.6 5.5 5.0 5.3

Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that thepopulation standard deviations are equal. -A researcher was interested in comparing the heights of women in two different countries. Independent simple random samples of 9 women from country A and 9 women from country B yielded the following heights (in inches). Country A Country B 64.1 65.3 66.4 60.2 61.7 61.7 62.0 65.8 67.3 61.0 64.9 64.6 64.7 60.0 68.0 65.4 63.6 59.0 Construct a $90 \%$ confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ , the difference between the mean height of women in country $\mathrm { A }$ and the mean height of women in country B. (Note: $\bar { x } _ { 1 } = 64.744$ in., $\bar { x } _ { 2 } = 62.556$ in., $s _ { 1 } = 2.192$ in., $\mathrm { s } _ { 2 } = 2.697$ in.)

Determine whether the samples are independent or dependent. -The accuracy of verbal responses is tested in an experiment in which individuals report their heights and then are measured. The data consist of the reported height and measured height for each individual.

Suppose that you wish to perform a traditional hypothesis test to test a claim regarding two means. Give an example of a situation in which the matched pairs test would be appropriate and give an example of a situation in which it would be appropriate to perform a test for large and independent samples.

Construct a confidence interval for µd, the mean of the differences d for the population of paired data. Assume that the population of paired differences is normally distributed. -A test of writing ability is given to a random sample of students before and after they completed a formal writing course. The results are given below. Construct a 99% confidence interval for the mean difference between the before and after scores. Before 70 80 92 99 93 97 76 63 68 71 74 After 69 79 90 96 91 95 75 64 62 64 76

Find the number of successes x suggested by the given statement. -A computer manufacturer randomly selects 2880 of its computers for quality assurance and finds that 2.43% of these computers are found to be defective.

Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. -Two types of flares are tested and their burning times are recorded. The summary statistics are given below. Brand X Brand Y =35 =40 =19.4 =15.1 =1.4 =0.8 Construct a 95\% confidence interval for the differences between the mean burning time of the brand $X$ flare and the mean burning time of the brand $Y$ flare.

Use the traditional method of hypothesis testing to test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations. -Five students took a math test before and after tutoring. Their scores were as follows. Subject A B C D E Before 76 68 68 70 71 After 80 77 66 73 83 Using a 0.01 level of significance, test the claim that the tutoring has an effect on the math scores.

Use the traditional method of hypothesis testing to test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations. -A test of abstract reasoning is given to a random sample of students before and after they completed a formal logic course. The results are given below. At the 0.05 significance level, test the claim that the mean score is not affected by the course. Before 74 83 75 88 84 63 93 84 91 77 After 73 77 70 77 74 67 95 83 84 75