Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.2.23

Chapter 8, Problem 8.2.23

Constructing Confidence Intervals for μ1-μ2, When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the population variances are not equal, you can construct a confidence interval for μ1-μ2 , as shown below.
construct the indicated confidence interval for μ1-μ2 . Assume the populations are approximately normal with unequal variances.
10K Race
To compare the mean finishing times of male and female participants in a 10K race, you randomly select several finishing times from both sexes. The results are shown at the left. Construct an 80% confidence interval for the difference in mean finishing times of male and female participants in the race. (Adapted from Great Race)

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Step 1: Identify the given statistics for both groups. For males: sample mean (x̄₁) = 65.8, sample standard deviation (s₁) = 34.1, and sample size (n₁) = 20. For females: sample mean (x̄₂) = 65.3, sample standard deviation (s₂) = 17.7, and sample size (n₂) = 18.

Step 2: Write the formula for the confidence interval for μ₁ - μ₂ when population variances are unequal. The formula is: CI = (x̄₁ - x̄₂) ± t* × √((s₁²/n₁) + (s₂²/n₂)), where t* is the critical t-value for the given confidence level and degrees of freedom.

Step 3: Calculate the degrees of freedom using the formula for unequal variances: df = ((s₁²/n₁ + s₂²/n₂)²) / (((s₁²/n₁)² / (n₁ - 1)) + ((s₂²/n₂)² / (n₂ - 1))). This will help determine the appropriate t* value for the 80% confidence level.

Step 4: Look up the critical t-value (t*) for the calculated degrees of freedom and the 80% confidence level. Use a t-distribution table or statistical software to find this value.

Step 5: Substitute the values for x̄₁, x̄₂, s₁, s₂, n₁, n₂, and t* into the confidence interval formula to compute the interval bounds. Ensure all calculations are performed step by step to avoid errors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, such as 80%, indicating the probability that the interval includes the parameter. In this case, it will estimate the difference in mean finishing times between male and female participants in the 10K race.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used when estimating population parameters when the sample size is small or when the population variance is unknown. In this scenario, the t-distribution is appropriate due to the unequal variances of the two groups being compared.

Unequal Variances

Unequal variances, also known as heteroscedasticity, occur when two or more groups have different levels of variability in their data. This is important in statistical analysis because it affects the validity of certain tests and confidence intervals. In this case, the sample variances of male and female finishing times differ significantly, necessitating the use of specific formulas for constructing the confidence interval.

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Difference in Means: Hypothesis Tests Example 1

Related Practice

Textbook Question

Constructing Confidence Intervals for p1-p2 You can construct a confidence interval for the difference between two population proportions p1-p2 by using the inequality below.

$({\hat{p}}_{1} - {\hat{p}}_{2}) - z_{c} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}} < p_{1} - p_{2} < ({\hat{p}}_{1} - {\hat{p}}_{2}) + z_{c} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}}$

In Exercises 23–26, construct the indicated confidence interval for p1-p2. Assume the samples are random and independent.

Students Planning to Study Visual and Performing Arts In a survey of 10,000 students taking the SAT, 7% were planning to study visual and performing arts in college. In another survey of 8000 students taken 10 years before, 8% were planning to study visual and performing arts in college. Construct a 95% confidence interval for p1-p2, where p1 is the proportion from the recent survey and p2 is the proportion from the survey taken 10 years ago. (Adapted from College Board)

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Textbook Question

What conditions are necessary to use the z-test for testing the difference between two population proportions?

237

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Textbook Question

Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) calculate d̄ and Sd, (d) find the standardized test statistic t, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Therapeutic Taping

A physical therapist claims that the use of a specific type of therapeutic tape reduces pain in patients with chronic tennis elbow. The table shows the pain levels on a scale of 0 to 10, where 0 is no pain and 10 is the worst pain possible, for 15 patients with chronic tennis elbow when holding a 1 kilogram weight. At , α=0.05 is there enough evidence to support the therapist’s claim? (Adapted from BioMed Central, Ltd.)

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Textbook Question

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.

Claim: μd≤0 , α=0.10, Sample statistics: d̄ =6.5, sd=9.54, n=16

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Textbook Question

Gas Mileage The table shows the gas mileages (in miles per gallon) of eight cars with and without using a fuel additive. At α=0.10, is there enough evidence to conclude that the additive improved gas mileage? Assume the populations are normally distributed.

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Textbook Question

A pediatrician claims that the mean birth weight of a single-birth baby is greater than the mean birth weight of a baby that has a twin. The mean birth weight of a random sample of 85 single-birth babies is 3086 grams. Assume the population standard deviation is 563 grams. The mean birth weight of a random sample of 68 babies that have a twin is 2263 grams. Assume the population standard deviation is 624 grams. At α=0.10, can you support the pediatrician’s claim? Interpret the decision in the context of the original claim.

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