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 8.6-8.2

Chapter 8, Problem 8.6-8.2

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.

Verified step by step guidance

Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) states that the fuel additive does not improve gas mileage (mean difference = 0). The alternative hypothesis (H₁) states that the fuel additive improves gas mileage (mean difference > 0).

Step 2: Calculate the differences between the gas mileage with and without the fuel additive for each car. For example, for Car 1, the difference is 23.6 - 23.1 = 0.5. Repeat this for all cars to obtain the differences.

Step 3: Compute the mean and standard deviation of the differences. Use the formulas for sample mean and sample standard deviation: mean = (Σ differences) / n, and standard deviation = sqrt(Σ(difference - mean)² / (n-1)), where n is the number of cars.

Step 4: Perform a one-sample t-test for the mean of the differences. Calculate the t-statistic using the formula: t = (mean difference) / (standard deviation / sqrt(n)).

Step 5: Compare the calculated t-statistic to the critical t-value at α = 0.10 for n-1 degrees of freedom. If the t-statistic exceeds the critical t-value, reject the null hypothesis and conclude that the additive improves gas mileage. Otherwise, fail to reject the null hypothesis.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. In this context, we formulate a null hypothesis (that the fuel additive does not improve gas mileage) and an alternative hypothesis (that it does). By analyzing the data, we can determine whether to reject the null hypothesis at a specified significance level, α.

Paired Sample t-Test

A paired sample t-test is used to compare two related groups, such as the gas mileage of the same cars before and after using a fuel additive. This test assesses whether the mean difference between the paired observations is significantly different from zero, allowing us to evaluate the effectiveness of the additive in improving gas mileage.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether the results of a hypothesis test are statistically significant. In this case, α=0.10 indicates a 10% risk of concluding that the fuel additive improves gas mileage when it does not. This level helps researchers decide how strong the evidence must be to reject the null hypothesis.

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Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.

μ≠2.28

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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.

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Textbook Question

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.

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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.

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Textbook Question

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Textbook Question

Describe another way you can perform a hypothesis test for the difference between the means of two populations using independent samples with and known that does not use rejection regions.

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