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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 7.4.11
Chapter 7, Problem 7.4.11

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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Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that more than half of all nurses feel they became better professionals during the pandemic. This translates to the alternative hypothesis Ha: p > 0.5, where p is the proportion of nurses who feel they became better professionals. The null hypothesis is the complement of this, H0: p = 0.5.
Step 2: Determine the critical value(s) and rejection region(s). Since the significance level is α = 0.01 and the test is one-tailed (right-tailed), use a z-table or statistical software to find the critical z-value corresponding to α = 0.01. The rejection region will be z > critical value.
Step 3: Calculate the standardized test statistic z. Use the formula z = (p̂ - p0) / √(p0(1 - p0) / n), where p̂ is the sample proportion (174/300), p0 is the hypothesized population proportion (0.5), and n is the sample size (300). Substitute the values into the formula to compute z.
Step 4: Compare the calculated z-value to the critical value. If the calculated z-value falls in the rejection region (z > critical value), reject the null hypothesis H0. Otherwise, fail to reject H0.
Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is enough evidence at the 0.01 significance level to support the manager's claim that more than half of all nurses feel they became better professionals during the pandemic. If the null hypothesis is not rejected, conclude that there is not enough evidence to support the claim.

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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 represents the claim being tested. The goal is to determine whether there is enough evidence in the sample to reject H0 in favor of Ha.
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Step 1: Write Hypotheses

Rejection Region

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level (α), which indicates the probability of making a Type I error (rejecting H0 when it is true). For a one-tailed test, like in this scenario, the rejection region is located in the tail of the distribution, where extreme values suggest that the null hypothesis is unlikely to be true.
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Step 4: State Conclusion

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an element is from the mean of the population. In hypothesis testing, it is calculated using the sample data and is compared against critical values to determine whether to reject the null hypothesis. A higher absolute value of z indicates that the sample result is further from the null hypothesis, providing stronger evidence against it.
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Step 2: Calculate Test Statistic
Related Practice
Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40

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 = 0.0062

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 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

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.

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.