Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 8 - Hypothesis Testing

Problem 8.1.27

Chapter 8, Problem 8.1.27

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.

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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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06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

Test Statistics

In Exercises 9–12, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 to select the correct expression for evaluating the test statistic.)

Exercise 5 “Landline Phones”

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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.

Online Friends A Pew Research Center poll of 1060 teens aged 13 to 17 showed that 57% of them have made new friends online. Use a 0.01 significance level to test the claim that half of all teens have made new friends online.

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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: 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.

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Textbook Question

Testing Claims About Variation

In Exercises 5–16, 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. Assume that a simple random sample is selected from a normally distributed population.

Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 g. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 g (based on Data Set 6 “Births” in Appendix B).

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Textbook Question

Exact Method For each of the three different methods of hypothesis testing (identified in the left column), enter the P-values corresponding to the given alternative hypothesis and sample data. Use a 0.05 significance level. Note that the entries in the last column correspond to the Chapter Problem. How do the results agree with the large sample size?

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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.

Landline Phones Claim: Fewer than 10% of homes have only a landline telephone and no wireless phone. Sample data: A survey by the National Center for Health Statistics showed that among 16,113 homes, 5.8% had landline phones without wireless phones.

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