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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 7.1.40
Chapter 7, Problem 7.1.40

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.


Lung Cancer A report claims that lung cancer accounts for 25% of all cancer diagnoses.

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1
Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ) in both words and symbols. The null hypothesis (H₀) states that the proportion of lung cancer diagnoses is equal to 25% (p = 0.25). The alternative hypothesis (Hₐ) depends on the context of the claim. If the claim is to test whether the proportion is different from 25%, then Hₐ: p ≠ 0.25. If the claim is to test whether the proportion is greater than or less than 25%, adjust Hₐ accordingly (e.g., Hₐ: p > 0.25 or Hₐ: p < 0.25).
Step 2: Determine the type of test (left-tailed, right-tailed, or two-tailed). If the alternative hypothesis is Hₐ: p ≠ 0.25, the test is two-tailed because we are testing for any difference from 25%. If Hₐ: p > 0.25, it is a right-tailed test, and if Hₐ: p < 0.25, it is a left-tailed test.
Step 3: Sketch the normal sampling distribution. Draw a bell-shaped curve representing the sampling distribution of the sample proportion under the null hypothesis. Mark the mean of the distribution at p = 0.25.
Step 4: Shade the area corresponding to the P-value. For a two-tailed test, shade both tails of the distribution beyond the critical values. For a left-tailed test, shade the area to the left of the test statistic. For a right-tailed test, shade the area to the right of the test statistic.
Step 5: Explain the reasoning. The type of test is determined by the alternative hypothesis. A two-tailed test is used when testing for any difference from the null hypothesis value, while a left-tailed or right-tailed test is used when testing for a specific direction of difference (less than or greater than the null hypothesis value).

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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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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

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


μ < 128

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 = 0.21

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.


Vaccinations In 2021, a reporter claims that at least 55% of U.S. adults feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. In a random sample of 200 U.S. adults, 56% feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. At α=0.10, is there enough evidence to reject the reporter’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