A researcher wants to determine whether there is a difference between two sunscreen lotions. Participants in amarathon race on a hot, sunny day applied lotion A to one arm and lotion B to the other arm. The results areshown in the table. $\begin{array}{cr|cc} & & \text { Lotion A } \\ & & \text { No burn (Success) } & \text { Burn (Failure) } \\ \hline & \text { No burn (Success) } & 720 & 41 \\ \text { Lotion B }\\ & \text { Burn (Failure) } & 51 & 74 \end{array}$ Is there a difference in the effectiveness of the two lotions in preventing sunburn? Use the critical-valuemethod and the α = 0.05 level of significance.

\[\begin{array} { l } \mathrm { H } _ {

Find the standardized test statistic, t, to test the hypothesis that μ1 = μ2. Two samples are randomly selectedand \[\begin{array} { l l } \mathrm { n } _ { 1 } = 14 & \mathrm { n } _ { 2 } = 12 \\ \overline { \mathrm { x } } 1 = 5 & \overline { \mathrm { x } } 2 = 6 \\ \mathrm {~s} _ { 1 } = 2.5 & \mathrm {~s} _ { 2 } = 2.8 \end{array}\] come from populations that are normal. The sample statistics are given below.

A) -0.954 B) -0.915 C) -1.558 D) -0.909 A) -0.954 B) -0.915 C) -1.558 D) -0.909

Test the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$. Two samples are randomly selected from normal populations. The sample statistics are given below. $\begin{array} { l l } \mathrm { n } _ { 1 } = 25 & \mathrm { n } _ { 2 } = 30 \\ \overline { \mathrm { x } } 1 = 22 & \overline { \mathrm { x } } 2 = 20 \\ \mathrm {~s} _ { 1 } = 1.5 & \mathrm {~s} _ { 2 } = 1.9 \end{array}$

critical value @#LAT-DLM& standardized t

\[\begin{array}{l} \text { Data sets A and B are dependent. Test the claim that } \mu _ { \mathrm { d } } = 0 \text {. Use } \alpha = 0.01 \text {. }\\ \begin{array} { c | c c c c c } \text { A } & 6.7 & 7.7 & 9.6 & 6.6 & 6.7 \\ \hline \text { B } & 9.1 & 8.0 & 7.9 & 7.8 & 9.2 \end{array} \end{array}\] Assume that the paired data came from a population that is normally distributed.

critical values @#LAT-DLM&; standardized

Construct a 95% confidence interval for data sets A and B. Data sets A and B are dependent. \[\begin{array} { c | l l l l l } \text { A } & 30 & 28 & 47 & 43 & 31 \\ \hline \text { B } & 28 & 24 & 25 & 35 & 22 \end{array}\] Assume that the paired data came from a population that is normally distributed.

A) (-0.696, 18.700) B) (-1.324, 8.981) C) (-0.113, 12.761) D) (-15.341, 15.431) A) (-0.696, 18.700) B) (-1.324, 8.981) C) (-0.113, 12.761) D) (-15.341, 15.431)

Exam 11: Inferences on Two Samples

A medical researcher suspects that the variance of the pulse rate of drinkers is higher than the variance of the pulse rate of non-drinkers. Use the sample statistics below to test the researcher's suspicion. Use $\alpha = 0.05 .$ Drinkers Non-drinkers $n _ { 1 } = 61$ $n _ { 2 } = 121$ $\mathrm { s } _ { 1 } = 5.46$ $s _ { 2 } = 3.71$ Assume the samples were randomly selected from normal populations.

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Question 1

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$\text { critical value } \mathrm { F } _ { 0 } = 1.43 \text {; test statistic } \mathrm { F } \approx 2.166 \text {; reject } \mathrm { H } _ { 0 } ; \text { There is sufficient evidence to support the hypothesis. }$

A researcher wishes to determine whether people with high blood pressure can lower their blood pressure by performing yoga exercises. A treatment group and a control group are selected. The sample statistics are given below. Construct a 90 % confidence interval for the difference between the two population means, $\mu _ { 1 } - \mu _ { 2 }$ Would you recommend using yoga exercises? Explain your reasoning. 1=100 =100 1=178 2=193 1=35 2=37

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confidence interval: $- 23.38 < \mu _ { 1 } = \mu _ { 2 } < - 6.623$ Since the interval does not contain zero, we can reject the claim of $\mu_ {1}$ = $\mu _ { 2 }$ Since the interval is negative, it appears that the yoga exercises lower blood pressure.

A survey found that 26 of 79 people who donʹt drink coffee often have trouble sleeping and 28 of 82 coffeedrinkers often have trouble sleeping. Construct a 95% confidence interval to determine whether there is adifference between the proportion of noncoffee drinkers who have trouble sleeping and the proportion ofcoffee drinkers who have trouble sleeping.

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Question 3

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$\mathrm { H } _ { 0 } : \mathrm { p } _ { 1 } = \mathrm { p } _ { 2 } \text { vs } \quad \mathrm { H } _ { 1 } : \mathrm { p } _ { 1 } \neq \mathrm { p } _ { 2 }$
Lower bound: $- 0.158$ , upper bound: $0.133$
Reject the null hypothesis. Because the confidence interval does not include zero, there is sufficient evidence at the $\alpha = 0.05$ level of significance to conclude that there is a difference in the proportions.

A researcher wants to determine whether there is a difference between two sunscreen lotions. Participants in amarathon race on a hot, sunny day applied lotion A to one arm and lotion B to the other arm. The results areshown in the table. Lotion A No burn (Success) Burn (Failure) No burn (Success) 720 41 Lotion B Burn (Failure) 51 74 Is there a difference in the effectiveness of the two lotions in preventing sunburn? Use the critical-valuemethod and the α = 0.05 level of significance.

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Question 4

Find the standardized test statistic, t, to test the hypothesis that μ1 = μ2. Two samples are randomly selectedand =14 =12 1=5 2=6 =2.5 =2.8 come from populations that are normal. The sample statistics are given below.

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Question 5

A local restaurant claims that the variance of waiting time for its patrons to be served is the lowest in the area.A competitor restaurant checks the waiting time at both restaurants. The sample statistics are listed below. Testthe local restaurantʹs hypothesis. Use α = 0.05. Local Restaurant Competitor Restaurant =41 =61 =1.43 minutes =2.6 minutes Assume the samples were randomly selected from normal populations.

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Question 6

To construct a confidence interval for the difference of two population proportions the samples must be independently obtained random samples, both must consist of less than $5 \%$ of the population, and

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Question 7

Test the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$ . Two samples are randomly selected from normal populations. The sample statistics are given below. =25 =30 1=22 2=20 =1.5 =1.9

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Question 8

A physician claims that a personʹs diastolic blood pressure can be lowered if, instead of taking a drug, theperson meditates each evening. Ten subjects are randomly selected and pretested. Their blood pressures,measured in millimeters of mercury, are listed below. The 10 patients are instructed in basic meditation andtold to practice it each evening for one month. At the end of the month, their blood pressures are taken again.The data are listed below. Test the physicianʹs claim. Use α = 0.01. Patient 1 2 3 4 5 6 7 8 9 10 Before 85 96 92 83 80 91 79 98 93 96 After 82 90 92 75 74 80 82 88 89 80

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Question 9

Data sets A and B are dependent. Test the claim that =0 . Use \alpha=0.01 . A 6.7 7.7 9.6 6.6 6.7 B 9.1 8.0 7.9 7.8 9.2 Assume that the paired data came from a population that is normally distributed.

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Question 10

Construct a 95% confidence interval for data sets A and B. Data sets A and B are dependent. A 30 28 47 43 31 B 28 24 25 35 22 Assume that the paired data came from a population that is normally distributed.

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Question 11

A random sample of 21 women had cholesterol levels with a variance of 1107.2. A random sample of 18 menhad cholesterol levels with a variance of 737.28. Test the hypothesis that the cholesterol levels for women havea larger variance than those for men. Use α = 0.01. Assume the samples were randomly selected from normalpopulations.

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Question 12

Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of thetwo procedures, random samples of the two types of surgical patients were obtained and the numbers ofpatients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown inthe table. How large a sample would be necessary in order to estimate the difference in the true success ratesto within Number of Successes Procedure A 100 73 Procedure B 100 92

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Question 13

What is the H₀ for testing differences of the means of two independent samples?

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Question 14

Construct a 98 % confidence interval for $P _ { 1 } - P _ { 2 }$ The sample statistics listed below are from independent samples. Sample statistics: $\mathrm { n } _ { 1 } = 1000 , \mathrm { x } _ { 1 } = 250 , \text { and } \mathrm { n } _ { 2 } = 1200 , \mathrm { x } _ { 2 } = 195$

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Question 15

In a recent survey of drinking laws, a random sample of 1000 women showed that 65 % were in favor of increasing the legal drinking age. In a random sample of 1000 men, 60 % favored increasing the legal drinking age. Construct a 95% confidence interval for $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$

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Question 16

Find the standardized test statistic, $\mathrm { z }$ to test the hypothesis that $\mathrm { p } _ { 1 } = \mathrm { p } _ { 2 }$ . Use $\alpha = 0.05$ . The sample statistics listed below are from independent samples. Sample statistics: $\mathrm { n } _ { 1 } = 50 , \mathrm { x } _ { 1 } = 35$ , and $\mathrm { n } _ { 2 } = 60 , \mathrm { x } _ { 2 } = 40$

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Question 17

A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Tensubjects are randomly selected and pretested. The results are listed below. The subjects were placed on thedrug for a period of 6 months, after which their cholesterol levels were tested again. The results are listedbelow. (All units are milligrams per deciliter.) Test the companyʹs claim that the drug lowers cholesterol levels.Use α = 0.01. Assume that the distribution is normally distributed. 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

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Question 18

Construct a 95% confidence interval for $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$ The sample statistics listed below are from independent samples. Sample statistics: $\mathrm { n } _ { 1 } = 50 , \mathrm { x } _ { 1 } = 35 , \text { and } \mathrm { n } _ { 2 } = 60 , \mathrm { x } _ { 2 } = 40$

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Question 19

A study was conducted to determine if the salaries of librarians from two neighboring cities were equal. Asample of 15 librarians from each city was randomly selected. The mean from the first city was $28,900 with astandard deviation of $2300. The mean from the second city was $30,300 with a standard deviation of $2100.Construct a 95% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$

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Question 20

Find the standardized test statistic, t, to test the hypothesis that μ1 = μ2. Two samples are randomly selectedand =14 =12 1=5 2=6 =2.5 =2.8 come from populations that are normal. The sample statistics are given below.

To construct a confidence interval for the difference of two population proportions the samples must be independently obtained random samples, both must consist of less than $5 \%$ of the population, and

Test the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$ . Two samples are randomly selected from normal populations. The sample statistics are given below. =25 =30 1=22 2=20 =1.5 =1.9

Data sets A and B are dependent. Test the claim that =0 . Use \alpha=0.01 . A 6.7 7.7 9.6 6.6 6.7 B 9.1 8.0 7.9 7.8 9.2 Assume that the paired data came from a population that is normally distributed.

Construct a 95% confidence interval for data sets A and B. Data sets A and B are dependent. A 30 28 47 43 31 B 28 24 25 35 22 Assume that the paired data came from a population that is normally distributed.

What is the H₀ for testing differences of the means of two independent samples?

Construct a 98 % confidence interval for $P _ { 1 } - P _ { 2 }$ The sample statistics listed below are from independent samples. Sample statistics: $\mathrm { n } _ { 1 } = 1000 , \mathrm { x } _ { 1 } = 250 , \text { and } \mathrm { n } _ { 2 } = 1200 , \mathrm { x } _ { 2 } = 195$

Data Collection

Organizing and Summarizing Data

Numerically Summarizing Data

Describing the Relation Between Two Variables

Probability

Discrete Probability Distributions

The Normal Probability Distribution

Sampling Distributions

Estimating the Value of a Parameter

Hypothesis Tests Regarding a Parameter

Inference on Categorical Data

Comparing Three or More Means

Filters

Exam 11: Inferences on Two Samples

Find the standardized test statistic, t, to test the hypothesis that μ1 = μ2. Two samples are randomly selectedand =14 =12 1=5 2=6 =2.5 =2.8 come from populations that are normal. The sample statistics are given below.

To construct a confidence interval for the difference of two population proportions the samples must be independently obtained random samples, both must consist of less than 5%5 \%5% of the population, and

Test the hypothesis that μ1=μ2\mu _ { 1 } = \mu _ { 2 }μ1​=μ2​ . Two samples are randomly selected from normal populations. The sample statistics are given below. =25 =30 1=22 2=20 =1.5 =1.9

Data sets A and B are dependent. Test the claim that =0 . Use \alpha=0.01 . A 6.7 7.7 9.6 6.6 6.7 B 9.1 8.0 7.9 7.8 9.2 Assume that the paired data came from a population that is normally distributed.

Construct a 95% confidence interval for data sets A and B. Data sets A and B are dependent. A 30 28 47 43 31 B 28 24 25 35 22 Assume that the paired data came from a population that is normally distributed.

What is the H0 for testing differences of the means of two independent samples?

Data Collection

Organizing and Summarizing Data

Numerically Summarizing Data

Describing the Relation Between Two Variables

Probability

Discrete Probability Distributions

The Normal Probability Distribution

Sampling Distributions

Estimating the Value of a Parameter

Hypothesis Tests Regarding a Parameter

Inference on Categorical Data

Comparing Three or More Means

Filters

To construct a confidence interval for the difference of two population proportions the samples must be independently obtained random samples, both must consist of less than $5 \%$ of the population, and

Test the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$ . Two samples are randomly selected from normal populations. The sample statistics are given below. =25 =30 1=22 2=20 =1.5 =1.9

What is the H₀ for testing differences of the means of two independent samples?