Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 8 - Hypothesis Testing

문제 8.1.16

8장, 문제 8.1.16

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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Step 1: Identify the type of hypothesis test. Since the claim is that p ≠ 0.455, this is a two-tailed test because the alternative hypothesis (H1) involves a 'not equal to' condition, which tests for deviations in both directions (greater than or less than).

Step 2: Determine the P-value. For a two-tailed test, the P-value is calculated as the sum of the probabilities in both tails of the standard normal distribution beyond the absolute value of the test statistic z = -1.60. Use the standard normal table or a statistical software to find the area to the left of z = -1.60, then double it to account for both tails.

Step 3: Compare the P-value to the significance level α = 0.05. If the P-value is less than or equal to 0.05, we reject the null hypothesis (H0). If the P-value is greater than 0.05, we fail to reject H0.

Step 4: Interpret the result. Based on the comparison in Step 3, decide whether there is sufficient evidence to support the claim that p ≠ 0.455 or if the evidence is insufficient to reject the null hypothesis.

Step 5: Conclude the hypothesis test. Clearly state whether H0 is rejected or not and provide a summary of the findings in the context of the problem, ensuring the decision aligns with the significance level and the calculated P-value.

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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (H1), which represents the effect or difference. The goal is to determine whether there is enough evidence in the sample to reject H0 in favor of H1.

P-Value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the assumption that H0 is true. A smaller P-value indicates stronger evidence against H0, and it is compared to a predetermined significance level (α) to decide whether to reject H0.

One-Tailed vs. Two-Tailed Tests

In hypothesis testing, tests can be classified as one-tailed or two-tailed based on the direction of the alternative hypothesis. A one-tailed test assesses the possibility of the relationship in one direction (either greater than or less than), while a two-tailed test evaluates both directions (not equal to). The choice between these tests affects the critical values and the interpretation of the P-value.

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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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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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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.

Internet Use A random sample of 5005 adults in the United States includes 751 who do not use the Internet (based on three Pew Research Center polls). Use a 0.05 significance level to test the claim that the percentage of U.S. adults who do not use the Internet is now less than 48%, which was the percentage in the year 2000. If there appears to be a difference, is it dramatic?

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Testing Claims About Proportions

Belief in Ghosts In a Harris Interactive poll of 2250 adults, 42% of the respondents said that they believe in ghosts. Use a 0.01 significance level to test the claim that more than of adults believe in ghosts.

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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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