Textbook Question
In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.
Sample 1: The retail prices of 20 motorcycles
Sample 2: The retail prices of 20 minivans
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Ch. 8 - Hypothesis Testing with Two Samples
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 8, Problem 8.R.22
In Exercises 19–22, 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=0.05.
Sample statistics: d̄=17.5, sd=4.05, n=37
Verified step by step guidance1
Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) is that the mean of the differences μd = 0, and the alternative hypothesis (H₁) is that μd ≠ 0. This is a two-tailed test since the claim is μd ≠ 0.
Step 2: Calculate the test statistic using the formula t = (d̄ - μd) / (sd / √n), where d̄ is the sample mean of the differences, μd is the hypothesized population mean of the differences (0 in this case), sd is the sample standard deviation of the differences, and n is the sample size.
Step 3: Determine the degrees of freedom (df) for the t-distribution. The degrees of freedom are calculated as df = n - 1, where n is the sample size.
Step 4: Find the critical t-value(s) for a two-tailed test at the significance level α = 0.05 and the calculated degrees of freedom. Use a t-distribution table or statistical software to find the critical values.
Step 5: Compare the calculated test statistic to the critical t-values. If the test statistic falls outside the range defined by the critical t-values, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Paired Data
Paired data refers to two sets of related observations, often collected from the same subjects under different conditions. This type of data is used in statistical tests to determine if there is a significant difference between the two conditions. In this context, the differences between paired observations are analyzed to test claims about their mean.
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Hypothesis Testing
Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0. In this case, the claim is that the mean of the differences (μd) is not equal to zero, which is tested against a significance level (α) of 0.05.
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06:21Step 1: Write Hypotheses
Significance Level (α)
The significance level, denoted as α, is the threshold for determining whether a statistical result is significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, an α of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none, guiding the decision-making process in hypothesis testing.
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Related Practice
Textbook Question
In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1> μ2; α=0.10. Assume (σ1)^2 ≠ (σ2)^2
Sample statistics: x̅1= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6
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Textbook Question
In Exercises 17 and 18, (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.
A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.
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Textbook Question
In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.
Sample 1: The heights of 37 children
Sample 2: The heights of the same 37 children after 1 year
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Textbook Question
In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1=μ2; α=0.01
Population statistics: σ1= 52 and σ2= 68
Sample statistics: x̅1 = 5595, n1 = 156, and x̅2= 5575, n2= 216
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Textbook Question
In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1>μ2; α=0.05
Population statistics: σ1= 0.30 and σ2= 0.23
Sample statistics: x̅1 = 1.28, n1 = 96, and x̅2= 1.34, n2= 85
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