9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

2:11 minutes

Problem 8.1.27

Textbook Question

Type I and Type II Errors
In Exercises 25–28, provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as p = 0.1.)

The proportion of drivers who make angry gestures is greater than 0.25.

Verified step by step guidance

Step 1: Understand the claim. The claim states that the proportion of drivers who make angry gestures is greater than 0.25. Symbolically, this can be written as H₁: p > 0.25, where p represents the true proportion of drivers who make angry gestures.

Step 2: Define the null hypothesis. The null hypothesis (H₀) is the opposite of the claim. It states that the proportion of drivers who make angry gestures is less than or equal to 0.25. Symbolically, H₀: p ≤ 0.25.

Step 3: Define a Type I error. A Type I error occurs when the null hypothesis is rejected even though it is true. In this context, a Type I error would mean concluding that the proportion of drivers who make angry gestures is greater than 0.25 (accepting H₁) when in reality, it is less than or equal to 0.25.

Step 4: Define a Type II error. A Type II error occurs when the null hypothesis is not rejected even though it is false. In this context, a Type II error would mean failing to conclude that the proportion of drivers who make angry gestures is greater than 0.25 (failing to accept H₁) when in reality, it is greater than 0.25.

Step 5: Summarize the errors. Type I error: Concluding p > 0.25 when p ≤ 0.25. Type II error: Failing to conclude p > 0.25 when p > 0.25.

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Key Concepts

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

Type I Error

A Type I error occurs when a true null hypothesis is incorrectly rejected. In the context of the given claim, it would mean concluding that the proportion of drivers who make angry gestures is greater than 0.25 when, in fact, it is not. This error represents a false positive, leading to the incorrect belief that there is an effect or difference when there is none.

Type II Error

A Type II error happens when a false null hypothesis is not rejected. For the claim regarding the proportion of drivers making angry gestures, this would mean failing to conclude that the proportion is greater than 0.25 when it actually is. This error represents a false negative, resulting in the missed opportunity to identify a true effect or difference.

Null Hypothesis and Alternative Hypothesis

In hypothesis testing, the null hypothesis (H0) represents a statement of no effect or no difference, while the alternative hypothesis (H1) suggests that there is an effect or difference. For the claim about drivers making angry gestures, the null hypothesis would state that the proportion is less than or equal to 0.25, while the alternative hypothesis would assert that it is greater than 0.25. Understanding these hypotheses is crucial for identifying Type I and Type II errors.

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Related Practice

Multiple Choice

In a certain hypothesis test, $H sub 0 colon p equals 0.4$ , $H_{a} : p$ < $0.4$ . You collect a sample and calculate a test statistic $z = - 1.32$ . Find the $P$ -value.

Stem Cell Survey In a Newsweek poll of 882 adults, 481 (or 55%) said that they were in favor of using federal tax money to fund medical research using stem cells obtained from human embryos. A politician claims that people don’t really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Use the following probabilities related to determining whether the result of 481 is significantly high (assuming the true rate is 50%). Is 481 significantly high? What should be concluded about the politician’s claim? Explain.

P(respondent says to use the federal tax money) = 0.5

P(among 882, exactly 481 says to use federal tax money) = 0.000713

P(among 882,481 or more say to use federal tax money) = 0.00389

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Textbook Question

Final Conclusions

In Exercises 21–24, use a significance level of α = 0.05 and use the given information for the following:

State a conclusion about the null hypothesis. (Reject H0 or fail to reject H0.)

Without using technical terms or symbols, state a final conclusion that addresses the original claim.

Original claim: The mean pulse rate (in beats per minute) of adult males is 72 bpm. The hypothesis test results in a P-value of 0.0095.

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Textbook Question

Type I and Type II Errors

In Exercises 25–28, provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as p = 0.1.)

The proportion of people who write with their left hand is equal to 0.1.

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Textbook Question

Interpreting Power Chantix (varenicline) tablets are used as an aid to help people stop smoking. In a clinical trial, 129 subjects were treated with Chantix twice a day for 12 weeks, and 16 subjects experienced abdominal pain (based on data from Pfizer, Inc.). If someone claims that more than 8% of Chantix users experience abdominal pain, that claim is supported with a hypothesis test conducted with a 0.05 significance level. Using 0.18 as an alternative value of p, the power of the test is 0.96. Interpret this value of the power of the test.

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Textbook Question

Identifying H0 and H1

In Exercises 5–8, do the following:

a. Express the original claim in symbolic form.

b. Identify the null and alternative hypotheses.

Light Year Claim: Most adults know that a light year is a measure of distance. Sample data: A Pew Research Center survey of 3278 adults showed that 72% knew that a light year is a measure of distance.

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Textbook Question

Identifying H0 and H1

In Exercises 5–8, do the following:

a. Express the original claim in symbolic form.

b. Identify the null and alternative hypotheses.

Systolic Blood Pressure Claim: Healthy adults have systolic blood pressure levels with a standard deviation greater than 5 mm Hg. Sample data: Data Set 1 “Body Data” in Appendix B shows that for 300 healthy adults, the systolic blood pressure amounts have a standard deviation of 15.85 mm Hg.

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Textbook Question

Statistical Literacy and Critical Thinking

In Exercises 1–4, use the results from a Hankook Tire Gauge Index survey of a simple random sample of 1020 adults. Among the 1020 respondents, 86% rated themselves as above average drivers. We want to test the claim that more than 3/4 of adults rate themselves as above average drivers.

Number and Proportions

b. Identify the sample proportion and use the symbol that represents it.

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