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Ch. 6 - Confidence Intervals
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. 6 - Confidence IntervalsProblem 6.17
Chapter 6, Problem 6.17

A researcher claims that 5% of people who wear eyeglasses purchase their eyeglasses online. Describe type I and type II errors for a hypothesis test of the claim. (Source: Consumer Reports)

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Understand the null hypothesis (H₀) and the alternative hypothesis (Hₐ): The null hypothesis (H₀) is that the proportion of people who purchase their eyeglasses online is 5% (p = 0.05). The alternative hypothesis (Hₐ) would be that the proportion is not 5% (p ≠ 0.05).
Define a Type I error: A Type I error occurs when the null hypothesis (H₀) is rejected even though it is true. In this context, it means concluding that the proportion of people who purchase their eyeglasses online is not 5% when, in reality, it is exactly 5%.
Define a Type II error: A Type II error occurs when the null hypothesis (H₀) is not rejected even though it is false. In this context, it means failing to conclude that the proportion of people who purchase their eyeglasses online is not 5% when, in reality, it is different from 5%.
Relate the errors to the researcher's claim: A Type I error would lead to an incorrect rejection of the researcher's claim that 5% of people purchase their eyeglasses online. A Type II error would lead to an incorrect acceptance of the researcher's claim when the true proportion is different from 5%.
Summarize the implications: Type I errors are associated with false alarms (rejecting a true claim), while Type II errors are associated with missed detections (failing to reject a false claim). Understanding these errors helps in interpreting the results of the hypothesis test and their potential consequences.

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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 inferences about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this context, the null hypothesis might state that 5% of eyeglass wearers purchase online, while the alternative could suggest a different proportion. The goal is to determine whether there is enough evidence to reject the null hypothesis.
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Step 1: Write Hypotheses

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. In this scenario, it would mean concluding that the proportion of eyeglass wearers who purchase online is different from 5% when, in fact, it is not. This error can lead to false claims about consumer behavior, potentially affecting marketing strategies and business decisions.
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Types of Data

Type II Error

A Type II error happens when the null hypothesis is not rejected when it is false. In this case, it would mean failing to recognize that the proportion of eyeglass wearers who buy online is different from 5%, leading to missed opportunities for businesses to adapt to actual consumer preferences. Understanding this error is crucial for evaluating the effectiveness of the hypothesis test and its implications.
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Related Practice
Textbook Question

Constructing Confidence Intervals In Exercises 25 and 26, use the figure, which shows the results of a survey in which 1051 adults from France, 1042 adults from Germany, 1003 adults from the United Kingdom, and 1000 adults from the United States were asked whether national identity is strongly tied to birthplace. (Source: Pew Research Center)

National Identity Construct a 99% confidence interval for the population proportion of adults who say national identity is strongly tied to birthplace for each country listed.

Textbook Question

In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.

(0.087, 0.263)

Textbook Question

Choosing a Distribution In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

Body Mass Index In a random sample of 50 people, the mean body mass index (BMI) was 27.7 and the standard deviation was 6.12.

Textbook Question

In Exercises 7 and 8, find the margin of error for the values of c, s, and n.

c = 0.95, s = 5, n = 16

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Textbook Question

Drug Concentration You are analyzing the times for the drug concentrations to peak in the patients in Exercise 14. The population standard deviation of the times for epinephrine concentrations to peak should be less than 10 minutes. Does the confidence interval you constructed for σ suggest that the variation in the times is at an acceptable level? Explain your reasoning.

Textbook Question

In Exercises 13–16, find the margin of error for the values of c, σ and n.

c = 0.95, σ = 5.2, n = 30