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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 7.4.4
Chapter 7, Problem 7.4.4

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p ≥0.48, α=0.08. Sample statistics: p_hat = 0.40, n=90

Verified step by step guidance
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Step 1: Verify the conditions for using a normal sampling distribution. The two conditions are: (1) The sample size n must be large enough such that both n * p and n * (1 - p) are greater than or equal to 5, and (2) the sampling must be random and independent.
Step 2: Calculate n * p and n * (1 - p) using the claimed population proportion p = 0.48 and sample size n = 90. Use the formulas: n * p and n * (1 - p).
Step 3: Check if both n * p and n * (1 - p) are greater than or equal to 5. If they are, then the normal approximation can be used. If not, the normal approximation cannot be used.
Step 4: If the normal approximation is valid, calculate the test statistic z using the formula: z = (p̂ - p) / sqrt((p * (1 - p)) / n), where p̂ is the sample proportion, p is the claimed proportion, and n is the sample size.
Step 5: Compare the calculated z-value to the critical z-value for the given significance level α = 0.08 in a one-tailed test. Determine whether to reject or fail to reject the null hypothesis based on this comparison.

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Key Concepts

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

Normal Sampling Distribution

A normal sampling distribution is applicable when the sample size is sufficiently large, typically n ≥ 30, and the population proportion is not too close to 0 or 1. This allows the sampling distribution of the sample proportion (p_hat) to be approximated by a normal distribution, facilitating hypothesis testing and confidence interval estimation.
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Sampling Distribution of Sample Proportion

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1, based on a predetermined significance level (α).
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Guided course
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Step 1: Write Hypotheses

Sample Proportion (p_hat)

The sample proportion (p_hat) is the ratio of the number of successes in a sample to the total number of observations in that sample. It serves as an estimate of the population proportion (p) and is crucial for conducting hypothesis tests regarding population proportions, especially when comparing it to a claimed value.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (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.


[APPLET] Fluorescent Lamps A compact fluorescent lamp (CFL) bulb manufacturer guarantees that the mean life of a CFL bulb is at least 10,000 hours. You want to test this guarantee. To do so, you record the lives of a random sample of 32 CFL bulbs. The results (in hours) are listed. Assume the population standard deviation is 1850 hours. At alpha=0.11, do you have enough evidence to reject the manufacturer’s claim?


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Textbook Question

In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.


Left-tailed test, α=0.10, n=20

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Textbook Question

What are the two types of hypotheses used in a hypothesis test? How are they related?

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Textbook Question

A travel analyst claims the mean daily base price for renting a full-size or less expensive vehicle in Vancouver, British Columbia, is more than \$86. You want to test this claim. In a random sample of 40 full-size or less expensive vehicles available to rent in Vancouver, British Columbia, the mean daily base price is \$93.23. Assume the population standard deviation is \$28.90. At α=0.10, do you have enough evidence to support the analyst’s claim?

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Textbook Question

A government agency reports that the mean amount of earnings for full-time workers ages 18 to 24 with a bachelor’s degree in a recent year is \$52,133. In a random sample of 15 full-time workers ages 18 to 24 with a bachelor’s degree, the mean amount of earnings is \$48,400 and the standard deviation is \$6679. At α=0.05, is there enough evidence to reject the claim? Assume the population is normally distributed.

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Textbook Question

[APPLET] A weight loss program claims that program participants have a mean weight loss of at least 10.5 pounds after 1 month. The weight losses after 1 month (in pounds) of a random sample of 40 program participants are listed below. At α=0.01, is there enough evidence to reject the program’s claim?


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