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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.RE.27
Chapter 7, Problem 7.RE.27

In Exercises 27 and 28, (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.


A substance abuse counselor claims that the mean annual drug overdose death rate for the 50 states is at least 25 deaths per 100,000 people. In a random sample of 30 states, the mean annual drug overdose rate is 22.48 per 100,000 people. Assume the population standard deviation is 10.69 deaths per 100,000. At α=0.01, is there enough evidence to reject the claim?

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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 mean annual drug overdose death rate for the 50 states is at least 25 deaths per 100,000 people. This translates to H0: μ ≥ 25 (the null hypothesis) and Ha: μ < 25 (the alternative hypothesis, which is the complement of the claim). This is a left-tailed test because the alternative hypothesis is testing for a value less than 25.
Step 2: Determine the critical value(s) and rejection region(s). Since this is a left-tailed test with a significance level of α = 0.01, use the standard normal (Z) distribution table to find the critical value corresponding to α = 0.01. The rejection region will be in the left tail of the Z-distribution, where Z is less than the critical value.
Step 3: Calculate the standardized test statistic (z). Use the formula for the Z-test statistic for a population mean: z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean (22.48), μ is the hypothesized population mean (25), σ is the population standard deviation (10.69), and n is the sample size (30). Substitute the given values into the formula to compute z.
Step 4: Compare the calculated test statistic (z) to the critical value. If the test statistic falls in the rejection region (i.e., if z is less than the 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 counselor's claim that the mean annual drug overdose death rate is at least 25 deaths per 100,000 people. If the null hypothesis is not rejected, conclude that there is not enough evidence to reject 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. In this case, H0 states that the mean annual drug overdose death rate is less than 25, while Ha claims it is at least 25.
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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 rejecting H0. For this problem, with α=0.01, the critical value will help identify if the sample mean significantly deviates from the hypothesized mean.
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Critical Values: t-Distribution

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an element is from the mean. It is calculated using the sample mean, population mean under H0, population standard deviation, and sample size. In this scenario, calculating the z-value will help determine if the observed sample mean of 22.48 is significantly lower than the hypothesized mean of 25, thus aiding in the decision to reject or fail to reject H0.
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Step 2: Calculate Test Statistic
Related Practice
Textbook Question

In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.


Left-tailed test, α=0.05, n=48

Textbook Question

In Exercises 7–10, explain how you should interpret a decision that rejects the null hypothesis.


A nonprofit consumer organization says that the standard deviation of the starting prices of its top-rated vehicles for a recent year is no more than \$2900.

Textbook Question

In Exercises 55–58, test the claim about the population variance or standard deviation at the level of significance . Assume the population is normally distributed.


Claim: σ^2 > 2; α=0.10. Sample statistics: s^2 = 2.95, n=18

Textbook Question

In Exercises 51–54, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance.


Left-tailed test, n=6, α=0.05

Textbook Question

In Exercises 7–10, describe type I and type II errors for a hypothesis test of the claim.


An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.

Textbook Question

In Exercises 7–10, (c) explain whether the hypothesis test is left-tailed, right-tailed, or two-tailed.


An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.