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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.10c
Chapter 7, Problem 7.RE.10c

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.

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Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis represents the claim to be tested, while the alternative hypothesis represents the claim the researcher wants to support. Here, H₀: μ ≥ 25 (the mean number of grams of carbohydrates is 25 or more) and Hₐ: μ < 25 (the mean number of grams of carbohydrates is less than 25).
Determine the direction of the inequality in the alternative hypothesis (Hₐ). Since the alternative hypothesis uses '<' (less than), this indicates a left-tailed test.
Understand the implications of a left-tailed test. In a left-tailed test, the critical region (where we would reject the null hypothesis) is located in the left tail of the sampling distribution.
Visualize the hypothesis test. Imagine a normal distribution curve with the critical region on the left side. This helps confirm that the test is left-tailed because we are testing for values significantly less than 25.
Conclude that the hypothesis test is left-tailed based on the structure of the alternative hypothesis and the direction of the inequality.

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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 a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis typically represents a statement of no effect or no difference, while the alternative hypothesis reflects the claim being tested. The outcome of the test determines whether to reject or fail to reject the null hypothesis.
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Step 1: Write Hypotheses

One-Tailed vs. Two-Tailed Tests

In hypothesis testing, a one-tailed test evaluates the direction of the effect, either greater than or less than a certain value, while a two-tailed test assesses whether there is a significant difference in either direction. A left-tailed test is used when the alternative hypothesis states that a parameter is less than a certain value, whereas a right-tailed test is used when it states that the parameter is greater. Understanding the directionality is crucial for correctly interpreting the results.
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Mean and Population Parameters

The mean is a measure of central tendency that represents the average of a set of values. In hypothesis testing, the population mean is often the parameter of interest, and claims about it are tested using sample data. In this context, the claim that the mean number of grams of carbohydrates in an energy bar is less than 25 grams indicates a specific population parameter that is being evaluated against the null hypothesis.
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Related Practice
Textbook Question

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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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

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Textbook Question

n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


μ ≤ 375

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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=15

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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

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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

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