Textbook Question
Writing A null hypothesis is rejected with a level of significance of 0.10. Is it also rejected at a level of significance of 0.05? Explain.
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 7.2.32
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
Verified step by step guidance1
Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The claim is μ ≤ 22,500, so the null hypothesis is H₀: μ ≤ 22,500, and the alternative hypothesis is Hₐ: μ > 22,500. This is a right-tailed test.
Step 2: Calculate the test statistic using the formula for a one-sample z-test: z = (x̄ - μ₀) / (σ / √n), where x̄ is the sample mean, μ₀ is the population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size.
Step 3: Substitute the given values into the formula. Here, x̄ = 23,500, μ₀ = 22,500, σ = 1200, and n = 45. Compute the standard error (SE) first: SE = σ / √n.
Step 4: Determine the critical value for a right-tailed test at α = 0.01. Use a z-table or statistical software to find the z-critical value corresponding to a significance level of 0.01.
Step 5: Compare the calculated z-test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. 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 is that the population mean is less than or equal to 22,500.
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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 0.01, indicating a 1% risk of concluding that the population mean exceeds 22,500 when it does not. This threshold helps determine the critical value for making decisions based on the sample data.
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04:46Step 4: State Conclusion Example 4
Sample Mean and Standard Error
The sample mean (x̄) is the average value calculated from the sample data, which in this case is 23,500. The standard error (SE) measures the variability of the sample mean and is calculated as the population standard deviation divided by the square root of the sample size (n). Understanding these concepts is crucial for calculating test statistics and making inferences about the population mean.
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05:11Sampling Distribution of Sample Proportion
Related Practice
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
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 ≠ 5880; α = 0.03; α = 413
Sample statistics: x_bar = 5771, n = 67
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.