Textbook Question
Describe the difference between calculating the standardized test statistic, Z^2, for a chi-square test for variance and a chi-square test for standard deviation.
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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 7.1.23
Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.
Ha: σ^2 = 142
H0: σ ≠ 142
Verified step by step guidance1
Step 1: Understand the null hypothesis (H0) and the alternative hypothesis (Ha). In this case, H0 states that the population variance (σ^2) is equal to 142, while Ha states that the population variance is not equal to 142.
Step 2: Recognize the type of comparison being made in the alternative hypothesis (Ha). If Ha uses '≠', it indicates that the test is looking for any difference, either greater than or less than the value specified in H0.
Step 3: Recall that a hypothesis test with '≠' in the alternative hypothesis is a two-tailed test because it considers deviations in both directions (greater than and less than the specified value).
Step 4: Confirm that the test is two-tailed by noting that the rejection regions for the null hypothesis will be in both tails of the distribution.
Step 5: Conclude that this hypothesis test is a two-tailed test based on the structure of the alternative hypothesis (Ha: σ^2 ≠ 142).
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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 two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.
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06:21
Step 1: Write Hypotheses
Types of Hypothesis Tests
Hypothesis tests can be classified as left-tailed, right-tailed, or two-tailed based on the direction of the alternative hypothesis. A left-tailed test is used when Ha suggests that a parameter is less than a certain value, while a right-tailed test is used when Ha suggests it is greater. A two-tailed test is appropriate when Ha indicates that the parameter is simply different from a specified value, without specifying a direction.
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06:21Step 1: Write Hypotheses
Variance and Standard Deviation
Variance (σ^2) is a measure of the dispersion of a set of data points around their mean, indicating how much the values differ from the average. In hypothesis testing, the population variance is often tested to understand the variability of the data. The null hypothesis in this case (H0: σ ≠ 142) suggests that the population variance is not equal to 142, while the alternative hypothesis (Ha: σ^2 = 142) posits that it is equal to this value.
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08:45Calculating Standard Deviation
Related Practice
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.
P= 0.2802
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Textbook Question
Deciding on a Distribution In Exercises 31 and 32, decide whether you should use the standard normal sampling distribution or a t-sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim.
Tuition and Fees An education publication claims that the mean in-state tuition and fees at public four-year institutions by state is more than \$10,500 per year. A random sample of 30 states has a mean in-state tuition and fees at public four-year institutions of \$10,931 per year. Assume the population standard deviation is \$2380. At α=0.01, test the publication’s claim.
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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.
Phone Repairs A cell phone repair shop advertises that the mean cost of repairing a phone screen is less than \$120.
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Textbook Question
Writing You are testing a claim and incorrectly use the standard normal sampling distribution instead of the t-sampling distribution, mistaking the sample standard deviation for the population standard deviation. Does this make it more or less likely to reject the null hypothesis? Is this result the same no matter whether the test is left-tailed, right-tailed, or two-tailed? Explain your reasoning.
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Textbook Question
The mean of a random sample of 18 test scores is x_bar. The standard deviation of the population of all test scores is sigma= 6. Under what condition can you use a z-test to decide whether to reject a claim that the population mean is mu=88?
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