Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 8 - Hypothesis Testing

문제 8.1.25

8장, 문제 8.1.25

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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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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Interpreting P-value The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analyzed with a formal hypothesis test with the alternative hypothesis of p > 0.5 which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is greater than 0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer as a result in your hypothesis test: 0.999, 0.5, 0.95, 0.05, 0.01, 0.001? Why?

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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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Minting Dollar Coins For the sample data from Exercise 1, we get a P-value of 0.0041 when testing the claim that σ < 0.04000 g.

What should we conclude about the null hypothesis?

What should we conclude about the original claim?

What do these results suggest about the new minting process?

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Finding P-Values

In Exercises 13–16, do the following:

i. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.

ii. Find the P-value. (See Figure 8-3.)

iii. Using a significance level of α = 0.05 should we reject H0 or should we fail to reject H0?

The test statistic of z = -0.75 is obtained when testing the claim that p<1/3.

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Finding P-Values

In Exercises 13–16, do the following:

i. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.

ii. Find the P-value. (See Figure 8-3.)

iii. Using a significance level of α = 0.05 should we reject H0 or should we fail to reject H0?

The test statistic of z = -1.60 is obtained when testing the claim that p ≠ 0.455.

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Randomization: Testing a Claim About a Mean

In Exercises 9–12, use the randomization procedure for the indicated exercise.

Section 8-3, Exercise 21 “Lead in Medicine”

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