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Ch. 7 - Hypothesis Testing with One Sample
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 7 - Hypothesis Testing with One Sample문제 7.5.30
7장, 문제 7.5.30

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 12 randomly chosen nursing supervisors are shown in the table at the left. At α=0.10, is there enough evidence to reject the claim that the standard deviation of the annual salaries is \$18,630?


tab2

검증된 단계별 안내
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Step 1: Identify the claim and state the null hypothesis (H0) and the alternative hypothesis (Ha). The claim is that the standard deviation of the annual salaries is \(18,630. Thus, H0: σ = 18,630 (the standard deviation equals \)18,630), and Ha: σ ≠ 18,630 (the standard deviation does not equal \$18,630).
Step 2: Determine the critical value(s) and rejection region(s). Since the test involves the standard deviation and the population is normally distributed, use the chi-square distribution. The degrees of freedom (df) are calculated as n - 1, where n is the sample size. Here, n = 12, so df = 11. Using α = 0.10 for a two-tailed test, find the critical chi-square values from a chi-square table or calculator.
Step 3: Calculate the standardized test statistic X². First, compute the sample variance (s²) using the formula s² = Σ(x - x̄)² / (n - 1), where x̄ is the sample mean. Then, use the formula X² = (n - 1) * s² / σ² to calculate the test statistic, where σ² is the square of the claimed standard deviation.
Step 4: Compare the test statistic X² to the critical values. If X² falls within the rejection region (either below the lower critical value or above the upper critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is enough evidence to reject the claim that the standard deviation of the annual salaries is \$18,630. If the null hypothesis is not rejected, conclude that there is not enough evidence to reject the claim.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.
추천 영상:
가이드 코스
06:21
Step 1: Write Hypotheses

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0. If the calculated test statistic falls within this region, we reject the null hypothesis.
추천 영상:
05:50
Critical Values: t-Distribution

Standard Deviation and Chi-Square Test

The standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of hypothesis testing for standard deviation, the Chi-Square test is used to determine if the sample standard deviation significantly differs from a hypothesized population standard deviation. The test statistic is calculated using the sample data, and its value is compared to the critical value to make a decision regarding the null hypothesis.
추천 영상:
가이드 코스
08:45
Calculating Standard Deviation
관련 실천
교과서 질문

Graphical Analysis In Exercises 57–60, you are given a null hypothesis and three confidence intervals that represent three samplings. Determine whether each confidence interval indicates that you should reject H0. Explain your reasoning.

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교과서 질문

Graphical Analysis In Exercises 57–60, you are given a null hypothesis and three confidence intervals that represent three samplings. Determine whether each confidence interval indicates that you should reject H0. Explain your reasoning.

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교과서 질문

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.


Lead Levels As part of your work for an environmental awareness group, you want to test a claim that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. You find that the mean amount of lead in the air for a random sample of 56 U.S. cities is 0.021 microgram per cubic meter and the standard deviation is 0.034 microgram per cubic meter. At α=0.01, can you support the claim?

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교과서 질문

Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.


Ha: μ ≤ 8.0

H0: μ > 8.0

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교과서 질문

Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


σ^2 ≥ 1.2

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교과서 질문

True or False? In Exercises 5–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


To support a claim, state it so that it becomes the null hypothesis.

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