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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: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 7 - Hypothesis Testing with One SampleProblem 7.1.39
Chapter 7, Problem 7.1.39

Identifying the Nature of a Hypothesis Test In Exercises 37–42, state and in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value.


Golf A golf analyst claims that the standard deviation of the 18-hole scores for a golfer is less than 2.1 strokes.

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁) in both words and symbols. The null hypothesis (H₀) represents the claim that the standard deviation of the 18-hole scores for the golfer is equal to or greater than 2.1 strokes. Symbolically, H₀: σ ≥ 2.1. The alternative hypothesis (H₁) represents the claim that the standard deviation is less than 2.1 strokes. Symbolically, H₁: σ < 2.1.
Step 2: Determine the type of hypothesis test. Since the alternative hypothesis (H₁) uses a 'less than' symbol (σ < 2.1), this indicates that the test is a left-tailed test. This is because we are testing whether the standard deviation is significantly smaller than the hypothesized value.
Step 3: Sketch the normal sampling distribution. Draw a bell-shaped curve to represent the sampling distribution of the test statistic. Mark the hypothesized value of the standard deviation (σ = 2.1) on the horizontal axis. Shade the left tail of the curve, as this represents the area corresponding to the P-value for a left-tailed test.
Step 4: Explain the reasoning for the tail direction. The left-tailed test is appropriate because the claim in the alternative hypothesis (H₁) is that the standard deviation is less than 2.1 strokes. The P-value will represent the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
Step 5: Prepare to calculate the test statistic and P-value. To proceed, you would use the chi-square test for variance or standard deviation, as the claim involves the standard deviation. The test statistic is calculated using the formula: χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s is the sample standard deviation, and σ₀ is the hypothesized standard deviation (2.1 strokes). The P-value is then determined using the chi-square distribution with (n - 1) degrees of freedom.

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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 (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.
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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 the alternative hypothesis states that a parameter is less than a certain value, while a right-tailed test is used when it states that the parameter is greater. A two-tailed test is appropriate when the alternative hypothesis indicates that the parameter is simply different from a certain value, without specifying a direction.
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06:21
Step 1: Write Hypotheses

P-value and Normal Distribution

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. In the context of a normal distribution, the P-value corresponds to the area under the curve in the tail(s) of the distribution, which is shaded to visually represent the likelihood of observing the sample data if the null hypothesis holds.
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Step 3: Get P-Value
Related Practice
Textbook Question

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.

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 7–12, (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 z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.


Nursing A patient care manager claims that more than half of all nurses feel they became better professionals during the coronavirus pandemic. In a random sample of 300 nurses, 174 say they became better professionals during the pandemic. At α=0.01, is there enough evidence to support the manager’s claim?

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Textbook Question

Hypothesis Testing Using a P-Value In Exercises 13–16, (a) identify the claim and state H0 and Ha, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim.


Stray Cats An animal advocate claims that 25% of U.S. households have taken in a stray cat. In a random sample of 500 U.S. households, 105 say they have taken in a stray cat. At α=0.05, is there enough evidence to reject the advocate’s claim?

Textbook Question

Graphical Analysis In Exercises 17–20, match the alternative hypothesis with its graph. Then state the null hypothesis and sketch its graph.


Ha: μ > 3


a.

b.

c.

d.

Textbook Question

In Exercise 1, you rejected the claim that p=0.53. But this claim was true. What type of error is this?

Textbook Question

Interpreting a P-Value In Exercises 3–8, the P-value for a hypothesis test is shown. Use the P-value to decide whether to reject H0 when the level of significance is (a)α=0.01, (b) α=0.05 , and (c) α=0.10.


P = 0.0838