9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

2:02 minutes

Problem 8.1.25

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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Understand the context: The problem involves hypothesis testing, where we are given a claim about the proportion of people who write with their left hand (p = 0.1). We need to identify the Type I and Type II errors associated with this claim.

Define the null and alternative hypotheses: The null hypothesis (H₀) is that the proportion of people who write with their left hand is equal to 0.1 (H₀: p = 0.1). The alternative hypothesis (H₁) is that the proportion of people who write with their left hand is not equal to 0.1 (H₁: p ≠ 0.1).

Recall the definition of a Type I error: A Type I error occurs when the null hypothesis (H₀) is true, but we incorrectly reject it. In this case, a Type I error would mean concluding that the proportion of left-handed people is not 0.1 when, in fact, it is 0.1.

Recall the definition of a Type II error: A Type II error occurs when the null hypothesis (H₀) is false, but we fail to reject it. In this case, a Type II error would mean failing to conclude that the proportion of left-handed people is not 0.1 when, in fact, it is different from 0.1.

Summarize the errors: A Type I error corresponds to rejecting H₀ (p = 0.1) when it is true, and a Type II error corresponds to failing to reject H₀ (p = 0.1) when it is false.

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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 left-handed people is not equal to 0.1 when, in fact, it is. This error is often referred to as a 'false positive' and is denoted by the significance level alpha (α), which represents the probability of making this error.

Type II Error

A Type II error happens when a false null hypothesis is not rejected. In relation to the claim about left-handedness, this would mean failing to conclude that the proportion of left-handed people is different from 0.1 when it actually is. This error is known as a 'false negative' and is represented by beta (β), which indicates the probability of making this error.

Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. In this scenario, the null hypothesis posits that the proportion of left-handed individuals is equal to 0.1. Understanding the null hypothesis is crucial for identifying Type I and Type II errors, as these errors are defined in relation to whether this hypothesis is correctly accepted or rejected.

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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.

Textbook Question

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

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: More than 35% of air travelers would choose another airline to have access to inflight Wi-Fi. The hypothesis test results in a P-value of 0.00001.

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.

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.

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.

Textbook Question

Testing Claims About Proportions

In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use a 0.01 significance level to test the claim that most medical malpractice lawsuits are dropped or dismissed. Should this be comforting to physicians?

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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