Textbook Question
In Exercises 13–18, test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed.
Claim: μ=4915; α=0.01. Sample statistics: x_bar=5017, s=5613, n=51
Ch. 7 - Hypothesis Testing with One Sample
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 7, Problem 7.1.60
Graphical Analysis In Exercises 57–60, you are given a null hypothesis and three confidence intervals that represent three samplings. Determine whether each confidence interval indicates that you should reject H0. Explain your reasoning.
Verified step by step guidance1
Step 1: Understand the null hypothesis (H0). The null hypothesis states that p ≥ 0.73, where p represents the population proportion. To reject H0, the confidence interval must not include values greater than or equal to 0.73.
Step 2: Analyze confidence interval (a). The interval is 0.73 < p < 0.75. Since this interval starts at 0.73 and includes values greater than 0.73, it does not provide evidence to reject H0.
Step 3: Analyze confidence interval (b). The interval is 0.715 < p < 0.725. This interval does not include 0.73 or any values greater than 0.73, which suggests that H0 can be rejected based on this sampling.
Step 4: Analyze confidence interval (c). The interval is 0.695 < p < 0.745. This interval includes 0.73, so it does not provide evidence to reject H0.
Step 5: Summarize the findings. Confidence interval (b) indicates rejection of H0 because it excludes 0.73 and values greater than 0.73, while intervals (a) and (c) do not provide evidence to reject H0 as they include values greater than or equal to 0.73.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Null Hypothesis (H0)
The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the default assumption in statistical testing. In this context, H0 states that the population proportion (p) is greater than or equal to 0.73. To determine whether to reject H0, we compare the confidence intervals from the samples to this hypothesis.
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Step 1: Write Hypotheses
Confidence Interval
A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter. Each confidence interval provides an estimate of where the true proportion (p) lies. If a confidence interval does not include the value specified in the null hypothesis, it suggests that we may reject H0.
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Statistical Significance
Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. In this scenario, if the confidence intervals for the sample proportions (p̂) do not overlap with the null hypothesis value (0.73), it indicates that the results are statistically significant, leading to a potential rejection of H0.
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Related Practice
Textbook Question
Graphical Analysis In Exercises 57–60, you are given a null hypothesis and three confidence intervals that represent three samplings. Determine whether each confidence interval indicates that you should reject H0. Explain your reasoning.
Textbook Question
Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.
Ha: μ ≤ 8.0
H0: μ > 8.0
Textbook Question
Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.
σ^2 ≥ 1.2
Textbook Question
In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.
Two-tailed test, n=61,α=0.01
Textbook Question
Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis test of the indicated claim.
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