Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.Q.5

Chapter 7, Problem 7.Q.5

A nonprofit consumer organization says that less than 25% of the televisions the organization rated in a recent year have an overall score of 70 or more. In a random sample of 35 televisions the organization rated in a recent year, 23% have an overall score of 70 or more. At α=0.05, can you support the organization’s claim?

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: p ≥ 0.25 (the proportion of televisions with a score of 70 or more is at least 25%), and the alternative hypothesis is Hₐ: p < 0.25 (the proportion of televisions with a score of 70 or more is less than 25%).

Step 2: Identify the sample proportion (p̂), sample size (n), and hypothesized population proportion (p₀). Here, p̂ = 0.23 (23%), n = 35, and p₀ = 0.25 (25%).

Step 3: Calculate the test statistic using the formula for a one-sample z-test for proportions: z = (p̂ - p₀) / √((p₀(1 - p₀)) / n). Substitute the values p̂ = 0.23, p₀ = 0.25, and n = 35 into the formula.

Step 4: Determine the critical value for a left-tailed test at α = 0.05. Use a z-table or statistical software to find the z-value corresponding to a cumulative probability of 0.05.

Step 5: Compare the calculated z-test statistic to the critical z-value. If the test statistic is less than the critical value, reject the null hypothesis (H₀) and conclude that there is sufficient evidence to support the organization's claim. Otherwise, fail to reject the null hypothesis.

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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). In this case, the null hypothesis would state that 25% or more of the televisions have a score of 70 or more, while the alternative hypothesis would assert that less than 25% do.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. In this scenario, α is set at 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none. This level helps to control the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected.

P-value

The p-value is a statistical measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of observing the sample data, or something more extreme, if the null hypothesis is true. If the p-value is less than the significance level (α), it suggests that the sample provides sufficient evidence to reject the null hypothesis in favor of the alternative.

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Step 3: Get P-Value

Related Practice

Textbook Question

A hat company claims that the mean hat size for a male is at least 7.25. A random sample of 12 hat sizes has a mean of 7.15. At α=0.01, can you reject the company’s claim? Assume the population is normally distributed and the population standard deviation is 0.27.

Textbook Question

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

μ ≠ 150,020

Textbook Question

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

μ = 82

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?

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.

Textbook Question

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

p ≥ 0.64