Exam 8: Hypothesis Testing With Two Samples

Suppose you want to test the claim that $\mu _ { 1 } < \mu _ { 2 }$ Assume the two samples are random and independent. At a level of significance of α $\alpha$ = 0.05, when should you reject H₀? Population statistics: $\sigma _ { 1 } = 2.9 \text { and } \sigma _ { 2 } = 2.8$ Sample statistics: $\overline { \mathrm { x } } 1 = 28.85 , \mathrm { n } _ { 1 } = 35 \text { and } \overline { \mathrm { x } } 2 = 31.4 , \mathrm { n } _ { 2 } = 42$

For which confidence interval for the difference in the means $\mu _ { 1 } - \mu _ { 2 } ,$ would you reject the null hypothesis?

Test the claim that $\mu _ { \mathrm { d } } \neq$ using the sample statistics below. Assume the samples are random and dependent, and the populations are normally distributed. Use $\alpha = 0.02$ Sample statistics: $\mathrm { n } = 13 , \overline { \mathrm { d } } = 1.1 , \mathrm {~s} _ { \mathrm { d } } = 5.3$

Suppose you want to test the claim that $\mu _ { 1 } > \mu _ { 2 }$ AssumAssume the two samples are random and independent. At a level of significance of α = 0.01, when should you reject H₀? Population statistics: $\sigma _ { 1 } = 45 \text { and } \sigma _ { 2 } = 25$ Sample statistics: $\overline { \mathrm { x } } 1 = 805 , \mathrm { n } _ { 1 } = 100 \text { and } \overline { \mathrm { x } } 2 = 790 , \mathrm { n } _ { 2 } = 125$

Two samples are random and independent. Find the P-value used to test the claim that $\mu _ { 1 } > \mu _ { 2 } . \text { Use } \alpha = 0.05 \text {. }$ Population statistics: $\sigma _ { 1 } = 40 \text { and } \sigma _ { 2 } = 24$ Sample statistics: $\overline { \mathrm { x } } 1 = 615 , \mathrm { n } _ { 1 } = 100 \text { and } \bar { x } _ { 2 } = 600 , \mathrm { n } _ { 2 } = 125$

In the initial test of the Salk vaccine for polio, 400,000 children were randomly selected and divided into two groups of 200,000. One group was vaccinated with the Salk vaccine while the second group was vaccinated with a placebo. Of those vaccinated with the Salk vaccine, 33 later developed polio. Of those receiving the placebo, 115 later developed polio. Test the claim that the Salk vaccine is effective in lowering the polio rate. Use $\alpha = 0.01$

A local bank claims that the waiting time for its customers to be served is the lowest in the area. A competitorʹs bank checks the waiting times at both banks. Assume the samples are random and independent, and the populations are normally distributed. Test the local bankʹs claim: (a)assuming that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text {, and (b) }$ assuming that $\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 } \text {. Use } \alpha = 0.05 \text {. }$ Local Bank =15 1=5.3 minutes =1.1 minute Competitor Bank =16 2=5.6 minutes =1.0 minutes

A medical researcher suspects that the pulse rate of smokers is higher than the pulse rate of non-smokers. Test the researcherʹs suspicion using α $\alpha$ = 0.05. Assume the two samples are random and independent. Smokers Nonsmokers =100 =100 1=87 2=84 =4.8 =5.3

Suppose you want to test the claim that $\mu _ { 1 } < \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _{ 2 } ^{ 2 } \text {. }$ At a Level of significance of α $\alpha = 0.10 \text {, }$ when should you reject H₀? =15 =15 =27.37 =29.92 =2.9 =2.8

A nutritionist believes that obesity is more prevalent among American adults than it was in the past. He discovers that in a study conducted in the year 1994, 380 of the 1630 randomly chosen adults were classified as obese. However, in a more recent study, he finds 726 out of 2350 randomly chosen adults were classified as obese. At α $\alpha$ = 0.05, do these studies provide evidence to support the nutritionistʹs claim that the proportion of obese adults has significantly increased since 1994?

A statistics teacher wanted to see whether there was a significant difference in ages between day students and night students. A sample of 35 students is selected from each group. The data are given below. Assume the two samples are random and independent. Test the claim that there is no difference in age between the two groups. Use α $\alpha = 0.05$ Day Students 22 24 24 23 19 19 23 22 18 21 21 18 18 25 29 24 23 22 22 21 20 20 20 27 17 19 18 21 20 23 26 30 25 21 25 Evening Students 18 23 25 23 21 21 23 24 27 31 24 20 20 23 19 25 24 27 23 20 20 21 25 24 23 28 20 19 23 24 20 27 21 29 30

A sports analyst claims that the mean batting average for teams in the American League is not equal to the mean batting average for teams in the National League because a pitcher does not bat in the American League. The data listed below are random, independent, and come from populations that are normally distributed. At $\alpha$ α = 0.05, test the sports analystʹs claim. Assume the population variances are equal. $\text { American League }$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { National League }$ 0.279 0.274 0.271 0.268 0.265 0.254 0.240 0.284 0.267 0.266 0.263 0.261 0.259 0.256

A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten subjects are randomly selected and pretested. The results are listed below. The subjects were placed on the drug for a period of 6 months, after which their cholesterol levels were tested again. The results are listed below. (All units are milligrams per deciliter.)Test the companyʹs claim that the drug lowers cholesterol levels. Assume the samples are random and dependent, and the populations are normally distributed. Use α $\alpha = 0.01 .$ Subject 1 2 3 4 5 6 7 8 9 10 Before 195 225 202 195 175 250 235 268 190 240 After 180 220 210 175 170 250 205 250 190 225

Construct a 95% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ Assume the two samples are random and independent. The sample statistics are given below. Population statistics: $\sigma _ { 1 } = 2.5 \text { and } \sigma _ { 2 } = 2.8$ Sample statistics: $\overline { \mathrm { x } } 1 = 12 , \mathrm { n } _ { 1 } = 40 \text { and } \overline { \mathrm { x } } 2 = 13 , \mathrm { n } _ { 2 } = 35$

Suppose you want to test the claim that $\mu _ { 1 } \neq \mu _ { 2 }$ Assume the two samples are random and independent. At a level of significance of α = 0.02, when should you reject H₀? Population statistics: $\sigma _ { 1 } = 0.76 \text { and } \sigma _ { 2 } = 0.51$ Sample statisticsstatistics: $\overline { \mathrm { x } } 1 = 1.8 , \mathrm { n } _ { 1 } = 51 \text { and } \overline { \mathrm { x } } 2 = 2.2 , \mathrm { n } _ { 2 } = 38$

Test the claim that $\mu _ { 1 } \neq \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that $\sigma _ { 1 } ^ { 2 } \neq \sigma_{ 2 }^ { 2 } \text {. Use } \alpha = 0.02$ =11 =18 =4 =4.4 =0.76 =0.51

Suppose you want to test the claim that $\mu _ { 1 } < \mu _ { 2 }$ Assume the two samples are random and independent. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis?

Test the claim that $\mu _ { 1 } = \mu _ { 2 }$ Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text {. }$ Use $\alpha$ = 0.05. =14 =12 =12 =13 =2.5 =2.8

Find the standardized test statistic to test the claim that $\mu _ { 1 } > \mu _ { 2 } .$ Assume the two samples are random and independent. Population statistics: $\sigma _ { 1 } = 45 \text { and } \sigma _ { 2 } = 25$ Sample statistics: $\bar { x } 1 = 480 , n _ { 1 } = 100 \text { and } \bar { x } { 2 } = 465 , n _ { 2 } = 125$

Find the weighted estimate, $\overline { \mathrm { p } }$ to test the claim that $\mathrm { p } _ { 1 } > \mathrm { p } _ { 2 } \text {. Use } \alpha = 0.01$ Assume the samples are random and independent. Sample statistics: $\mathrm { n } _ { 1 } = 100 , \mathrm { x } _ { 1 } = 38 , \text { and } \mathrm { n } _ { 2 } = 140 , \mathrm { x } _ { 2 } = 50$