Exam 9: Inferences From Two Samples

Assume that you plan to use a significance level of $\alpha = 0.05$ to test the claim that $p _ { 1 } = p _ { 2 }$ Use the given sample sizes and numbers of successes to find the P -value for the hypothesis test. =50 =75 =20 =15

Suppose you wish to test a claim about the mean of the differences from dependent samples or_ to construct a confidence interval estimate of the mean of the differences from dependent samples. What are the requirements?

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

Express the alternative hypothesis in symbolic form. An automobile technician claims that the mean amount of time (in hours)per domestic car repair is more than that of foreign cars. Assume that two samples are independent. Let the domestic car repair times be the first Population and the foreign car repair times be the second population.

Test the indicated claim about the means of two populations. 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. A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure, measured in mm Hg, by following a particular diet. Use a significance level of 0.01 to test the claim that the treatment group is from a population with a smaller mean than the control group. Include your null and alternative hypotheses, the test statistic, P-value or critical value(s), conclusion about the null hypothesis, and conclusion about the claim in your answer. Treatment Group Control Group =35 =75 =189.1 =203.7 =38.7 =39.2

Find the number of successes x suggested by the given statement. A computer manufacturer_ randomly selects 2680 of its computers for quality assurance and finds that 1.98% of these Computers are found to be defective.

Brian wants to obtain a confidence interval estimate of $p _ { 1 } - p _ { 2 }$ where $p _ { 1 }$ represents the proportion of American women who smoke and $p _ { 2 }$ represents the proportion of American men who smoke. He randomly selects 100 married couples. Among the 100 women in the sample are 21 smokers. Among the 100 men are 29 smokers. Are the requirements for obtaining a confidence interval estimate of $p _ { 1 } - p _ { 2 }$ satisfied? If not, which requirement is not satisfied?

A researcher wishes to compare how students at two different schools perform on a math test._ He randomly selects 40 students from each school and obtains their test scores. He pairs the first score from school A with the first school from school B, the second score from school A with the second school from school B and so on. He then performs a hypothesis test for matched pairs. Is this approach valid? Why or why not? If it is not valid, how should the researcher have proceeded?

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. $\text { t-Test: Two Sample for Means }$ 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 t -1.773417 7 P(T<=t) one-tail 0.0384 8 T Critical one-tail 1.644853 9 P(T<=t) two tail 0.0768 10 t Critical two-tail 1.959961

When testing the claim that $p _ { 1 } = p _ { 2 }$ , a test statistic of z=2.04 is obtained. Find the P -value obtained from this test statistic.

Test the indicated claim about the variances or standard deviations of two populations. Assume that both samples are independent simple random samples from populations having normal distributions. A random sample of 16 women resulted in blood pressure levels with a standard deviation of 23 mm Hg. A random sample of 17 men resulted in blood pressure levels with a standard deviation of 19.2 mm Hg. Use a 0.05 significance level to test the claim that blood pressure levels for women vary more than blood pressure levels for men. Include your null and alternative hypotheses, the test statistic, P-value or critical value(s), conclusion about the null hypothesis, and conclusion about the claim in your answer.

Assume that two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Which distribution is used to test the claim that women have a higher mean resting heart rate Than men?

Assume marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. At the 0.05 significance level, test the claim that the recognition rates are the same in both states. Include your null and alternative hypotheses, the test statistic, P-value or critical value(s), conclusion about the null hypothesis, and conclusion about the claim in your answer.

A paint manufacturer made a modification to a paint to speed up its drying time. Independent_ simple random samples of 11 cans of type A (the original paint)and 9 cans of type B (the Modified paint)were selected and applied to similar surfaces. The drying times, in hours, were Recorded. The summary statistics are as follows. Type A Type B =76.3 =65.1 =4.5 =5.1 =11 =9 The following $98 \%$ confidence interval was obtained for $\mu _ { 1 } - \mu _ { 2 }$ , the difference between the mean drying time for paint cans of type $A$ and the mean drying time for paint cans of type $B$ : $4.90$ hrs $< \mu _ { 1 } - \mu _ { 2 } < 17.50$ hrs What does the confidence interval suggest about the population means? A) The confidence interval includes only positive values which suggests that the two population means might be equal. There doesn't appear to be a significant difference between the mean drying time for paint type A and the mean drying time for paint type B. The modification does not seem to be effective in reducing drying times. B) The confidence interval includes 0 which suggests that the two population means might be equal. There doesn't appear to be a significant difference between the mean drying time for paint type A and the mean drying time for paint type B. The modification does not seem to be effective in reducing drying times. C) The confidence interval includes only positive values which suggests that the mean drying time for paint type $\mathrm { A }$ is smaller than the mean drying time for paint type $\mathrm { B }$ . The modification does not seem to be effective in reducing drying times. D) The confidence interval includes only positive values which suggests that the mean drying time for paint type A is greater than the mean drying time for paint type B. The modification seems to be effective in reducing drying times.

To test the null hypothesis that the difference between two population proportions is equal to a nonzero constant c, use the test statistic $z = \frac { \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) - c } { \sqrt { \hat { p } _ { 1 } \left( 1 - \hat { p } _ { 1 } \right) / n _ { 1 } + \hat { p } _ { 2 } \left( 1 - \hat { p } _ { 2 } \right) / n _ { 2 } } }$ As long as $n _ { 1 }$ and $n _ { 2 }$ 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: $n _ { 1 } = 250 , x _ { 1 } = 146$ Women: $n _ { 2 } = 202 , x _ { 2 } = 103$

The two data sets are dependent. Find $\bar { d }$ to the nearest tenth. A 69 66 61 63 51 B 25 23 20 25 22

Test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations. Subject A B C D E Before 71 66 67 77 75 After 75 75 65 80 87 Using a 0.01 level of significance, test the claim that the tutoring has an effect on the math scores. Include your null and alternative hypotheses, the test statistic, P-value or critical value(s), conclusion about the null hypothesis, and conclusion about the claim in your answer.

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. A paint manufacturer wished to compare the drying times of two different types of Paint .Independent simple random samples of 11 cans of type A and 9 cans of type B were Selected and applied to similar surfaces. The drying times, in hours, were recorded. The Summary statistics are as follows. Type A Type B =75.7 =64.3 =4.5 =5.1 =11 =9 Construct a $99 \%$ confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ , the difference between the mean drying time for paint type A and the mean drying time for paint type $\mathrm { B }$ .

A random sample of 10 employees of an engineering company was selected. Each employee was asked to report the number of sick days he/she claimed on Wednesdays and Fridays of the previous calendar year. Use this information to test the employer's claim that more employees call in sick on Fridays than on Wednesdays. $\text { Use } \alpha = 0.05$ Assume that the differences between Wednesday's and Friday's sick day counts is normally distributed. Wednesdays Fridays 1 2 1 3 0 1 2 1 1 6 3 2 0 4 1 0 4 5 2 2 Include your null and alternative hypotheses, the test statistic, P-value or critical value(s), conclusion about the null hypothesis, and conclusion about the claim in your answer.

Construct a confidence interval for $\mu _ { d }$ 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