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 ≠ 5
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 7.3.15
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≥8000; α=0.01. Sample statistics: x_bar=77,000, s=450, n=25
Verified step by step guidance1
Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The claim is μ ≥ 8000, so the null hypothesis is H₀: μ ≥ 8000, and the alternative hypothesis is Hₐ: μ < 8000. This is a left-tailed test.
Step 2: Calculate the test statistic using the formula for a t-test: t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Substitute the given values: x̄ = 77000, μ₀ = 8000, s = 450, and n = 25.
Step 3: Determine the critical value for the t-distribution at the significance level α = 0.01 with degrees of freedom (df) = n - 1 = 25 - 1 = 24. Use a t-table or statistical software to find the critical value for a left-tailed test.
Step 4: Compare the calculated test statistic to the critical value. If the test statistic is less than the critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀.
Step 5: State the conclusion in the context of the problem. Based on the comparison in Step 4, determine whether there is sufficient evidence to reject the claim that μ ≥ 8000 at the 0.01 significance level.
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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 be that the population mean μ is less than 8000, while the alternative hypothesis would assert that μ is greater than or equal to 8000.
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06:21
Step 1: Write Hypotheses
Level of Significance (α)
The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that the population mean is greater than or equal to 8000 when it is not.
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04:46Step 4: State Conclusion Example 4
Sample Statistics
Sample statistics are numerical values calculated from a sample that provide insights into the population from which the sample is drawn. In this question, the sample mean (x̄ = 77,000), sample standard deviation (s = 450), and sample size (n = 25) are crucial for conducting the hypothesis test. These statistics will be used to calculate the test statistic and determine whether the evidence supports the claim about the population mean.
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05:11Sampling Distribution of Sample Proportion
Related Practice
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.
p < 0.45
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=81,α=0.10
Textbook Question
Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.
z = -2.37
Textbook Question
Hypothesis Testing Using a P-Value In Exercises 33–38,
a. identify the claim and state and .
b. find the standardized test statistic z.
c. find the corresponding P-value.
d. decide whether to reject or fail to reject the null hypothesis.
e. interpret the decision in the context of the original claim.
MCAT Scores A random sample of 100 medical school applicants at a university has a mean total score of 505 on the MCAT. According to a report, the mean total score for the school’s applicants is more than 503. Assume the population standard deviation is 10.6. At alpha=0.01, is there enough evidence to support the report’s claim?
1
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Textbook Question
Stating the Null and Alternative Hypotheses In Exercises 25–30, write the claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim.
Attendance An amusement park claims that the mean daily attendance at the park is at least 20,000 people.