Exam 11: Two-Sample Hypothesis Tests

The following display from a TI-84 Plus calculator presents the results of a hypothesis test for the difference between two proportions. The sample sizes are $n _ { 1 } = 87 \text { and } n _ { 2 } = 97$ Is this a left-tailed test, a right-tailed test, or a two-tailed test?

A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 } < \mu _ { 2 }$ . The sample means were $\bar { x } _ { 1 } = 8$ and $\bar { x } _ { 2 } = 14$ , the sample standard deviations were $s _ { 1 } = 6$ and $s _ { 2 } = 8$ , and the sample sizes were $n _ { 1 } = 12$ and $n _ { 2 } = 11$ . Is $H _ { 0 }$ rejected at the 0.05 level? (Hint: First compute the value of the test statistic.)

In an experiment to determine whether there is a systematic difference between the weights obtained with two different mass balances, six specimens were weighed, in grams, on each balance. The following data were obtained: Can you conclude that the mean weight differs between the two balances? i). State the null and alternative hypotheses. ii). Compute the test statistic. iii). State a conclusion using the $a = 0.05$ level of significance.

Following is a sample of five matched pairs. Sample 1 23 20 19 22 22 Sample 2 19 17 20 20 15 Let $\mu _ { 1 }$ and $\mu _ { 2 }$ represent the population means and let $\mu _ { \mathrm { d } } = \mu _ { 1 } - \mu _ { 2 }$ . A test will be made of the hypotheses $H _ { 0 } : \mu _ { \mathrm { d } } = 0$ versus $H _ { 1 } : \mu _ { \mathrm { d } } > 0$ . Can you reject $H _ { 0 }$ at the $\alpha = 0.05$ level of significance?

The following display from a TI-84 Plus calculator presents the results of a hypothesis test for the difference between two means. The sample sizes are $n _ { 1 } = 7 \text { and } n _ { 2 } = 13$ What is the P-value?

In a test for the difference between two proportions, the sample sizes were $n _ { 1 } = 104$ and $n _ { 2 } = 73$ , and the numbers of events were $x _ { 1 } = 52$ and $x _ { 2 } = 40$ . A test is made of the hypothesis $H _ { 0 } : p _ { 1 } = p _ { 2 }$ versus $H _ { 1 } : p _ { 1 } \neq p _ { 2 }$ . Can you reject $H _ { 0 }$ rejected at the $\alpha = 0.05$ level?

The following MINITAB output display presents the results of a hypothesis test for the difference $\mu _ { 1 } - \mu _ { 2 }$ between two population means. Two-sample T for X1 vs X2 Mean StDev SE Mean A 7 145.411 24.669 9.324 B 14 132.964 25.604 6.843 Difference $=m u(X 1)-m u(X 2)$ Estimate for difference: $12.447$ $95 \%$ CI for difference: $(-10.222,35.116)$ T-Test of difference $=0($ vs not $=)$ : $\quad$ T-Value $=1.076209$ $\text { P-Value }=0.301402 \quad \text { DF }=13$ What is the alternate hypothesis?

The following MINITAB output display presents the results of a hypothesis test on the difference between two proportions. Test and CI for Two Proportions: P1, P2 Variable X N Sample P P1 58 110 0.527273 P2 26 76 0.342105 Difference $= p ( \mathrm { P } 1 ) = p ( \mathrm { P } 2 )$ Estimate for difference:0.185168 $95 \%$ CI for difference:( $( 0.039677,0.330659 )$ T-Test of difference $= 0 ($ vs not $= 0 ):Z = 2.494513 \quad P$ -Value $= 0.012613$ What is the P-value?

A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 } < \mu _ { 2 }$ . The sample means were $\bar { x } _ { 1 } = 12$ and $\bar { x } _ { 2 } = 6$ , the sample standard deviations were $s _ { 1 } = 5$ and $s _ { 2 } = 4$ , and the sample sizes were $n _ { 1 } = 11$ and $n _ { 2 } = 19$ . Compute the value of the test statistic.

The following MINITAB output display presents the results of a hypothesis test on the difference between two proportions. Test and CI for Two Proportions: P1, P2 Variable X N Sample P P1 30 72 0.41666667 P2 36 97 0.37113402 Difference $= p ( \mathrm { P } 1 ) = p ( \mathrm { P } 2 )$ Estimate for differenc:0.04553265 95\% CI for difference:(-0.10321537, 0.19428066) T-Test of difference $= 0 ($ vs not $= 0 ) :Z =0.59996757 \quad P$ -Value $= 0.54852775$ Can you reject $H _ { 0 }$ rejected at the $\alpha = 0.05$ level?

The football coach at State University wishes to determine if there is a decrease in offensive production between the first half and the second half of his team's recent games. The table below shows the first-half and second-half offensive production (measured in total yards gained per half) for the past six games. Can you conclude that the mean offensive production in the first half differed from that of the seconc i). State the null and alternative hypotheses. ii). Compute the test statistic. iii). State a conclusion using the $\alpha$ =0.02 level of significance.

A broth used to manufacture a pharmaceutical product has its sugar content, in milligrams per milliliter, measured several times on two successive days. The results are shown below. Can you conclude that the variability of the process is greater on the second day than on the first day? Use the $\alpha = 0.01$ level of significance.

The following MINITAB output display presents the results of a hypothesis test for the difference $\mu _ { 1 } - \mu _ { 2 }$ between two population means. Two-sample T for X1 vs X2 A N Mean StDev SE Mean B 13 60.492 26.689 7.402 12 86.830 26.378 7.615 Difference $=m u(X 1)-m u(X 2)$ Estimate for difference: $-26.338$ $95 \%$ CI for difference: $(-47.152,-5.524)$ T-Test of difference $=0($ vs not $=)$ : T-Value $=-2.480133$ $\text { P-Value }=1.979112 \quad \text { DF }=23$ $\text { Can you reject } H _ { 0 } \text { rejected at the } \alpha = 0.10 \text { level? }$

The football coach at State University wishes to determine if there is a decrease in offensive production between the first half and the second half of his team's recent games. The table below Shows the first-half and second-half offensive production (measured in total yards gained per half) For the past six games. Compute the test statistic.

An automobile manufacturer wishes to test that claim that synthetic motor oil can improve gas mileage (in miles per gallon, or mpg). The table below shows the gas mileages, in mpg, of six cars That used synthetic motor oil. The table also shows the gas mileages in mpg of six cars that were Using conventional motor oil (the controls). Synthetic: 26 27 25 27 28 28 Control: 26 25 25 26 27 25 Can you conclude that the mean gas mileage for cars using synthetic motor oil is more than The mean for the controls? Use the $\alpha = 0.05$ level of significance.

In an experiment to determine whether there is a systematic difference between the weights obtained with two different mass balances, six specimens were weighed, in grams, on each balance. The following data Were obtained: State a conclusion using the $\alpha = 0.05$ level of significance.

An $F$ -test with 10 degrees of freedom in the numerator and 5 degrees of freedom in the denominator produced a test statistic whose value was $10.07$ . The null and alternate hypotheses were $H _ { 0 } : \sigma _ { 1 } = \sigma _ { 2 }$ versus $H _ { 1 } : \sigma _ { 1 } < \sigma _ { 2 }$ . Do you reject $H _ { 0 }$ at the $\alpha = 0.01$ level?

An amateur golfer wishes to determine if there is a difference between the drive distances of her two favorite drivers. (A driver is a specialized club for driving the golf ball down range.) She hits fourteen balls With driver A and 10 balls with driver B. The drive distances (in yards) for the trials are show below. Assume that the populations are approximately normal. Can you conclude that there is a difference in the Mean drive distances for the two drivers? Use the $\alpha = 0.05$ level of significance.

Are low-fat diets or low-carb diets more effective for weight loss? A sample of 70 subjects went on a low-carbohydrate diet for six months. At the end of that time, the sample mean weight loss was 10.5 pounds with a sample standard deviation of 7.09 pounds. A second sample of 76 subjects went on a low-fat diet. Their sample mean weight loss was 18.0 with a standard deviation of 7.26. Can you conclude that the mean weight loss differed between the two diets? Use the $\alpha = 0.05$ level. i). State the appropriate null and alternate hypotheses. ii). Compute the test statistic. iii). How many degrees of freedom are there, using the simple method? iv). Do you reject H₀ ? State a conclusion.

A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 } < \mu _ { 2 }$ . The sample means were $\bar { x } _ { 1 } = 14$ and $\bar { x } _ { 2 } = 15$ , the sample standard deviations were $s _ { 1 } = 7$ and $s _ { 2 } = 6$ , and the sample sizes were $n _ { 1 } = 20$ and $n _ { 2 } = 16$ . How many degrees of freedom are there for the test statistic, using the simple method?