Exam 9: Inferences From Two Samples

Test the indicated claim about the means of two populations. Assume that the two samples are independent and that they have been randomly selected. -Two types of flares are tested for their burning times (in minutes)and sample results are given below. Brand Brand n=35 =40 =19.4 =15.1 =1.4 =0.8 Refer to the sample data to test the claim that the two populations have equal means. Use a 0.05 significance level.

Assume that you want to test the claim that the paired sample data come from a population for which the mean difference is $\mu _ { \mathrm { d } } = 0$ . Compute the value of the $t$ test statistic. - 12.0 11.3 10.1 12.9 11.6 13.2 12.6 13.5 10.7 12.4

Use the computer display to solve the problem. -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 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 tCritical two-tail 1.959961

Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent and that they have been randomly selected. -A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Use the sample data below to construct a $99 \%$ confidence interval for $\mathrm { u } _ { 1 }$ - $\mathrm { u } _ { 2 }$ where $u _ { 1 }$ and $u _ { 2 }$ represent the mean for the treatment group and the control group respectively. Treatment Group Control Group =85 =75 =189.1 =203.7 =38.7 =39.2

Find the appropriate p-value to test the null hypothesis, H₀: p₁ = p₂, using a significance level of 0.05. - =200 =100 =11 =8

Solve the problem. -To test the null hypothesis that the difference between two population proportions is equal to a nonzero constant c, use the test statistic $z = \frac { \left. \hat { \left( \mathrm { p } _ { 1 } \right. } - \hat { \mathrm { p } _ { 2 } } \right) - \mathrm { c } } { \sqrt { \hat { \mathrm { p } } _ { 1 } \left( 1 - \hat { \mathrm { p } _ { 1 } } \right) / \mathrm { n } _ { 1 } + \hat { \mathrm { p } } _ { 2 } \left( 1 - \hat { \mathrm { p } } _ { 2 } \right) / \mathrm { n } _ { 2 } } }$ As long as $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ are both large, the sampling distribution of the test statistic $\mathrm { z }$ 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 10 percentage points more than the percentage of female voters who plan to vote Republican. Use the traditional method of hypothesis testing and use a significance level of $0.05$ . Men: $\mathrm { n } _ { 1 } = 250 , \mathrm { x } _ { 1 } = 146$ Women: $\mathrm { n } _ { 2 } = 202 , \mathrm { x } _ { 2 } = 103$

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 coach uses a new technique to train gymnasts. 7 gymnasts were randomly selected and their competition scores were recorded before and after the training. The results are shown below. Subject A B C D E F G Before 9.4 9.5 9.7 9.4 9.5 9.7 9.6 After 9.5 9.7 9.7 9.3 9.6 10 9.4 Using a 0.01 level of significance, test the claim that the training technique is effective in raising the gymnasts' scores.

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

Assume that you want to test the claim that the paired sample data come from a population for which the mean difference is $\mu _ { \mathrm { d } } = 0$ . Compute the value of the $t$ test statistic. - x 11 6 12 6 11 y 8 8 8 7 6

Solve the problem. -Two types of flares are tested for their burning times (in minutes)and sample results are given below. Use a 0.05 significance level to test the claim that the two brands have equal variances. Brand X Brand Y n=35 n =40 =19.4 =15.1 =1.4 =0.8

Provide an appropriate response. -Complete the table to describe each symbol. Identify the symbol Give the value or describe meaning how to find the value n

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal -Among 620 adults selected randomly from among the residents of one town, 25% said that they favor stronger gun-control laws.

Solve the problem. -The $95 \%$ confidence interval for a collection of paired sample data is $0.0 < \mu _ { \mathrm { d } } < 3.4$ . Based on the same sample, a traditional hypothesis test fails to support the claim of $\mathrm { u } _ { \mathrm { d } } > 0$ . What can you conclude about the significance level of the hypothesis test?

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 > . Use a significance level of 0.01 . Sample 1 Sample 2 =85 =90 =38 =23

Find the appropriate p-value to test the null hypothesis, H₀: p₁ = p₂, using a significance level of 0.05. - =100 =100 =38 =40

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal - =100 =100 =36 =37

Compute the test statistic used to test the null hypothesis that p₁ = p₂ -Information about movie ticket sales was printed in a movie magazine. Out of fifty PG-rated movies, 45% had ticket sales in excess of $3,000,000. Out of thirty-five R-rated movies, 27% grossed over $3,000,000.

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. -The table below shows the weights of 9 subjects before and after following a particular diet for two months. Subject A B C D E F G H I Before 168 180 157 132 202 124 190 210 171 After 162 178 145 125 171 126 180 195 163 Construct a $99 \%$ confidence interval for the mean difference of the "before" minus "after" weights.

Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected -In a random sample of 360 women, 65% favored stricter gun control laws. In a random sample of 220 men, 60% favored stricter gun control laws. Test the claim that the proportion of women favoring stricter gun control is higher than the proportion of men favoring stricter gun control. Use a significance level of 0.05.

Compute the test statistic used to test the null hypothesis that p₁ = p₂ -In a vote on the Clean Water bill, 45% of the 205 Democrats voted for the bill while 52% of the 230 Republicans voted for it.