Exam 9: Inferences From Two Samples

Provide an appropriate response. -Describe the process for testing hypothesis about two means when the samples are independent and large. Compare this process to the methods of hypothesis testing for one mean in Chapter 7.

Provide an appropriate response. -Define independent and dependent samples and give an example of each.

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. -Ten different families are tested for the number of gallons of water a day they use before and after viewing a conservation video. At the 0.05 significance level, test the claim that the mean is the same before and after the viewing. Before 33 33 38 33 35 35 40 40 40 31 After 34 28 25 28 35 33 31 28 35 33

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. - 7.1 5.9 3.4 10.3 4 10.2 9.3 7.1 5.3 5.6 4.5 5.6 4.4 4.9 5.4 5.4

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 0.10. Sample 1 Sample 2 =462 =380 =84 =95

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal -A computer manufacturer randomly selects 2140 of its computers for quality assurance and finds that 2.76% of these computers are found to be defective.

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal - =100 =100 =0.12 =0.1

Solve the problem. -When performing a hypothesis test for the ratio of two population variances, the upper critical F value is denoted $\mathrm { F } _ { \mathrm { R } }$ . The lower critical $\mathrm { F }$ value, $\mathrm { F } _ { \mathrm { L } }$ , can be found as follows: interchange the degrees of freedom, and then take the reciprocal of the resulting $\mathrm { F }$ value found in table $\mathrm { A } - 5 . \mathrm { F } _ { \mathrm { R } }$ can be denoted $\mathrm { F } _ { \alpha / 2 }$ and $\mathrm { F } _ { \mathrm { L } }$ can be denoted $\mathrm { F } _ { 1 } - \alpha / 2$ Find the critical values $\mathrm { F } _ { \mathrm { L } }$ and $\mathrm { F } _ { \mathrm { R } }$ for a two-tailed hypothesis test based on the following values: $\mathrm { n } _ { 1 } = 25 , \mathrm { n } _ { 2 } = 16 , \alpha = 0.10$

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 -A farmer has decided to use a new additive to grow his crops. He divided his farm into 10 plots and kept records of the corn yield (in bushels)before and after using the additive. The results are shown below. Plot: 1 2 3 4 5 6 7 8 9 10 Before 9 9 8 7 6 8 5 9 10 11 After 10 9 9 8 7 10 6 10 10 12 You wish the test the following hypothesis at the 1 percent level of significance. $\mathrm { H } _ { 0 } : \mu _ { \mathrm { D } } = 0$ against $\mathrm { H } _ { 1 } : \mu _ { \mathrm { D } } > 0$ . What decision rule would you use?

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 0.04 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 different from the mean for the placebo population? Explain. t-Test: Two Sample for Means 1 Variable 1 Variable 2 2 Mean 171.6392 168.7718 3 Known Variance 47.51672 41.08293 4 Observations 50 50 5 Hypothesized Mean Difference 0 6 t 2.154057 7 P(T>=t) one-tail 0.0158 8 TCritical one-tail 1.644853 9 P(T>=t) two-tail 0.0316 10 tCritical two-tail 1.959961

Provide an appropriate response. -Discuss the assumptions for constructing confidence intervals or hypothesis testing for two means from dependent samples.

Solve the problem. -When performing a hypothesis test for the ratio of two population variances, the upper critical $\mathrm { F }$ value is denoted $F _ { R }$ . The lower critical $F _ { \text {value, } } F _ { L }$ , can be found as follows: interchange the degrees of freedom, and then take the reciprocal of the resulting $\mathrm { F }$ value found in table A-5. F $\mathrm { F } _ { \mathrm { R } }$ can be denoted $\mathrm { F } _ { \alpha / 2 }$ and $\mathrm { F } _ { \mathrm { L } }$ can be denoted $\mathrm { F } _ { 1 } - \alpha / 2$ . Find the critical values $F _ { L }$ and $F _ { R }$ for a two-tailed hypothesis test based on the following values: $\mathrm { n } _ { 1 } = 9 , \mathrm { n } _ { 2 } = 7 , \alpha = 0.05$

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 500 people aged 20-24, 22% were smokers. In a random sample of 450 people aged 25-29, 14% were smokers. Test the claim that the proportion of smokers in the two age groups is the same. Use a significance level of 0.01.

Determine whether the samples are independent or consist of matched pairs. -The effectiveness of a new headache medicine is tested by measuring the amount of time before the headache is cured for patients who use the medicine and another group of patients who use a placebo drug.

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal - =44 =490 =17 =191

Solve the problem. -Test the claim that populations A and B have different variances. Use a significance level of 0.10. Sample A Sample B =28 =41 =19.2 =23.7 =4.7 =5.89

Test the indicated claim about the means of two populations. 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 test the claim that the treatment population mean µ1 is smaller than the control population mean µ2. Test the claim using a significance level of 0.01. Treatment Group Control Group =85 =75 =189.1 =203.7 =38.7 =39.2

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 -The following table shows the weights of nine subjects before and after following a particular diet for two months. You wish to test the claim that the diet is effective in helping people lose weight. What is the value of the appropriate test statistic? 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

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 -We wish to compare the means of two populations using paired observations. Suppose that $\bar { d } = 3.125 , \mathrm {~S} _ { \mathrm { d } } =$ $2.911$ , and $\mathrm { n } = 8$ , and that you wish to test the following hypothesis at the 5 percent level of significance: $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 0 \text { against } \mathrm { H } _ { 1 } : \boldsymbol { \mu } _ { \mathrm { d } } > 0 \text {. }$ What decision rule would you use?

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 30 38 25 26 28 27 33 35 y 28 34 34 26 29 32 33 34