Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.5.5

Chapter 7, Problem 7.5.5

How do the requirements for a chi-square test for a variance or standard deviation differ from a z-test or a t-test for a mean?

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Understand the purpose of the chi-square test: The chi-square test for variance or standard deviation is used to test hypotheses about the variability of a population, whereas z-tests and t-tests are used to test hypotheses about the population mean.

Identify the assumptions for the chi-square test: The chi-square test requires that the population being tested is normally distributed. This is a stricter assumption compared to z-tests and t-tests, which can sometimes be applied to approximately normal distributions or large sample sizes.

Recognize the difference in test statistics: The chi-square test statistic is calculated using the formula $ \chi^2 = \frac{(n-1)s^2}{\sigma^2} $, where $ n $ is the sample size, $ s^2 $ is the sample variance, and $ \sigma^2 $ is the hypothesized population variance. In contrast, z-tests and t-tests use formulas involving the sample mean and standard error.

Understand the degrees of freedom: For the chi-square test, the degrees of freedom are $ n-1 $, where $ n $ is the sample size. For t-tests, the degrees of freedom depend on the sample size and the type of test (e.g., one-sample or two-sample). Z-tests do not involve degrees of freedom as they are based on the standard normal distribution.

Note the type of hypothesis tested: The chi-square test is specifically designed to test hypotheses about variance or standard deviation, while z-tests and t-tests focus on hypotheses about the mean. This makes the chi-square test suitable for questions about variability rather than central tendency.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant difference between the expected and observed frequencies in categorical data. It is particularly useful for assessing variance or standard deviation in a population, requiring that the data be independent, randomly sampled, and that the sample size is sufficiently large to ensure the validity of the test.

Z-Test and T-Test

Z-tests and t-tests are statistical tests used to determine if there is a significant difference between the means of two groups. A z-test is appropriate when the population variance is known and the sample size is large (typically n > 30), while a t-test is used when the population variance is unknown and the sample size is smaller, relying on the t-distribution for more accurate results.

Assumptions of Tests

Each statistical test has specific assumptions that must be met for the results to be valid. For chi-square tests, the data should be categorical, while z-tests and t-tests assume that the data is continuous and normally distributed. Additionally, z-tests require known population variance, whereas t-tests do not, highlighting the differences in their application based on the nature of the data.

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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

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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?

105

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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.

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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=7,α=0.01

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Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 19–26, (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 t, (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.

Annual Salary An employment information service claims the mean annual salary for senior level statisticians is more than \$124,000. The annual salaries (in dollars) for a random sample of 12 senior level statisticians are shown in the table at the left. At α=0.01, is there enough evidence to support the claim that the mean salary is more than \$124,000?

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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≠52,200; α=0.05. Sample statistics: x_bar=53,220, s=2700, n=34

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