Textbook Question
In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225
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 7.2.31
In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.
Claim: ; μ ≠ 5880; α = 0.03; α = 413
Sample statistics: x_bar = 5771, n = 67
Verified step by step guidance1
Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: μ = 5880, and the alternative hypothesis is Hₐ: μ ≠ 5880. This is a two-tailed test because the claim specifies 'not equal to' (≠).
Step 2: Calculate the test statistic using the formula for a t-test: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Substitute the given values: x̄ = 5771, μ = 5880, s = 413, and n = 67.
Step 3: Determine the degrees of freedom (df) for the t-distribution. For a single-sample t-test, df = n - 1. In this case, df = 67 - 1 = 66.
Step 4: Find the critical t-value(s) for a two-tailed test at the significance level α = 0.03 and df = 66. Use a t-distribution table or statistical software to find the critical values. These values will define the rejection region for the null hypothesis.
Step 5: Compare the calculated t-value from Step 2 to the critical t-values from Step 4. If the calculated t-value falls in the rejection region (i.e., beyond the critical 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.
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about a population 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 favor of H1. In this case, the null hypothesis would state that the population mean μ equals 5880, while the alternative hypothesis claims that μ does not equal 5880.
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06:21
Step 1: Write Hypotheses
Level of Significance (α)
The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.03, indicating a 3% risk of concluding that a difference exists when there is none. This threshold helps determine the critical value for the test statistic, which is essential for making a decision regarding the null hypothesis.
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04:46Step 4: State Conclusion Example 4
Sample Mean and Standard Error
The sample mean (x̄) is the average value of a sample, which in this case is 5771. The standard error (SE) measures the dispersion of the sample mean from the population mean and is calculated using the sample standard deviation divided by the square root of the sample size (n). Understanding the sample mean and standard error is crucial for calculating the test statistic and determining whether the sample provides enough evidence to reject the null hypothesis.
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05:11Sampling Distribution of Sample Proportion
Related Practice
Textbook Question
Hypothesis Testing Using Rejection Regions In Exercises 23–30, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic X^2, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.
Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At α=0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056?
Textbook Question
In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.
Claim: ; μ ≤ 22,500; α = 0.01; α = 1200
Sample statistics: x_bar = 23,500, n = 45
Textbook Question
In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.
Claim: σ<40, α=0.01 . Sample statistics: s=40.8, n=12
Textbook Question
Explain how to test a population variance or a population standard deviation.
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.
Left-tailed test, n=24,α=0.05