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. =100 =100 =38 =40

Assume that you plan to use a significance level of $\alpha = 0.05$ to 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 =50 =8 =7

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.

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. Independent samples from two different populations yield the following data. $\bar { x } _ { 1 } = 958 , \bar { x } _ { 2 } = 157 , s _ { 1 } = 77 , s _ { 2 } = 88$ The sample size is 478 for both samples. Find the 85% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$

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?

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 mothers spend more time (in minutes) Driving their kids to activities than fathers do?

Assume that the following confidence interval for the difference in the mean length of male (sample 1) and female babies (sample 2) at birth was constructed using independent simple random samples. $- 0.2 \mathrm { in } < \mu _ { 1 } - \mu _ { 2 } < 2.7 \mathrm { in }$ What does the confidence interval suggest about the difference in length between male babies and female babies?

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.

Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis. Suppose you wish to test the claim that $\mu _ { d }$ the mean value of the differences d for a population of paired data, is greater than 0 . Given a sample of n=15 and a significance level of $\alpha = 0.01$ what criterion would be used for rejecting the null hypothesis?

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. Also assume that the population standard deviations are equal $\left( \sigma _ { 1 } = \sigma _ { 2 } \right)$ , so that the standard error of the difference between means is obtained by pooling the sample variances. A paint manufacturer wanted 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 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 B.

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: $n _ { 2 } = 202 , x _ { 2 } = 103$

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.

Assume that you want to test the claim that the paired sample data come from a population for which the mean difference is $\mu _ { d } = 0$ Compute the value of the t test statistics. Round intermediate calculations to four decimal places as needed and final answers to three decimal places as needed. x 9 6 7 5 12 y 6 8 3 6 7

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 $\mathrm { A }$ and the mean drying time for paint cans of type B: $4.90 \mathrm { hrs } < \mu _ { 1 } - \mu _ { 2 } < 17.50 \mathrm { hrs }$ What does the confidence interval suggest about the population means?

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. Using the sample paired data below, construct a 90% confidence interval for the population mean of all differences. 5.0 5.1 4.6 3.5 6.0 4.7 3.9 4.2 4.2 3.7

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.

Find $S _ { d }$ The differences between two sets of dependent data are 0.4,0.24,0.22,0.26,0.34 . Round to the nearest hundredth.

Test the indicated claim about the variances or standard deviations of two populations. that both samples are independent simple random samples from populations having normal distributions. When 25 randomly selected customers enter any one of several waiting lines, their waiting times have a standard deviation of 5.35 minutes. When 16 randomly selected customers enter a single main waiting line, their waiting times have a standard deviation of 2.2 minutes. Use a 0.05 significance level to test the claim that there is more variation in the waiting times when several lines are used. 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.

Which distribution is used to test the claim that the standard deviation of the lengths (in cm) of male babies at birth is equal to the standard deviation of the lengths (in cm)of female Babies at birth?