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Ch. 8 - Hypothesis Testing
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Triola 14th EditionCh. 8 - Hypothesis Testing문제 8.5.9
8장, 문제 8.5.9

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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Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). For this problem, the null hypothesis typically states that the mean lead content in the medicine is equal to a specified value (e.g., μ = μ₀), while the alternative hypothesis states that the mean lead content is different (e.g., μ ≠ μ₀).
Determine the test statistic to be used. Since this is a test about a mean, the test statistic is usually calculated as: x̄ − μ0 s n , where x̄ is the sample mean, μ0 is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
Simulate the randomization distribution. To do this, repeatedly shuffle or resample the data under the assumption that the null hypothesis is true. For each resample, calculate the test statistic (e.g., the mean). This will create a distribution of test statistics under the null hypothesis.
Compare the observed test statistic to the randomization distribution. Determine the proportion of simulated test statistics that are as extreme as or more extreme than the observed test statistic. This proportion represents the p-value.
Make a decision based on the p-value. If the p-value is less than the significance level (e.g., α = 0.05), reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem, explaining whether there is evidence to support the claim about the mean lead content in the medicine.

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주요 개념

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Randomization

Randomization is a statistical technique used to assign subjects to different groups in a way that eliminates bias. It ensures that each participant has an equal chance of being placed in any group, which helps to create comparable groups and allows for valid inferences about the effects of treatments or interventions. This method is crucial in hypothesis testing, particularly when assessing claims about population means.
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가이드 코스
07:09
Intro to Random Variables & Probability Distributions

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. It involves calculating a test statistic from sample data and comparing it to a critical value or using a p-value to assess the strength of the evidence. This process is essential for making informed decisions based on sample data regarding population parameters, such as means.
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가이드 코스
06:21
Step 1: Write Hypotheses

Mean

The mean, often referred to as the average, is a measure of central tendency that summarizes a set of values by dividing the sum of those values by the number of observations. In the context of hypothesis testing, the mean is used to represent the expected value of a population, and testing claims about the mean involves comparing sample means to determine if observed differences are statistically significant.
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Calculating the Mean
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교과서 질문

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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교과서 질문

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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교과서 질문

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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교과서 질문

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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교과서 질문

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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교과서 질문

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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