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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 8 - Hypothesis Testing with Two Samples문제 8.3.9f
8장, 문제 8.3.9f

Testing the Difference Between Two Means (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.
[APPLET] Migraines
A researcher claims that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The table shows the number of days chronic migraine sufferers suffered migraines before and after using the treatment. At , α= 0.01 is there enough evidence to support the researcher’s claim? (Adapted from Journal of Headache and Pain)
Table comparing the number of migraine days before and after treatment for chronic migraine patients.

검증된 단계별 안내
1
Step 1: Identify the type of test to use. Since the data involves paired samples (before and after treatment for the same patients), this is a dependent samples t-test (also known as a paired t-test). The goal is to test if the mean difference in migraine days before and after treatment is significantly less than zero, indicating a reduction in migraine days.
Step 2: State the null and alternative hypotheses. The null hypothesis (H₀) is that the mean difference in migraine days (before - after) is zero: H₀: μ_d = 0. The alternative hypothesis (Hₐ) is that the mean difference is less than zero: Hₐ: μ_d < 0.
Step 3: Calculate the differences for each patient. Subtract the 'Days after' values from the 'Days before' values to get the differences. For example, for Patient 1, the difference is 20 - 0 = 20. Repeat this for all patients to create a list of differences.
Step 4: Compute the test statistic. First, calculate the mean of the differences (d̄), the standard deviation of the differences (s_d), and the number of paired observations (n). Then, use the formula for the t-statistic: t = (d̄ - 0) / (s_d / √n).
Step 5: Compare the test statistic to the critical value. Using a t-distribution table and α = 0.01 with degrees of freedom df = n - 1, find the critical value for a one-tailed test. If the calculated t-statistic is less than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Dependent Samples

Dependent samples, also known as paired samples, occur when the same subjects are measured before and after a treatment. This design is crucial for analyzing the effect of an intervention, as it controls for individual variability. In this case, the number of migraine days before and after treatment for the same patients allows for a direct comparison of the treatment's effectiveness.
추천 영상:
05:17
Multiplication Rule: Dependent Events

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to support a specific claim about a population parameter. In this scenario, the null hypothesis would state that there is no difference in the number of migraine days before and after treatment, while the alternative hypothesis posits that the treatment does reduce the number of days. The significance level (α = 0.01) indicates the threshold for rejecting the null hypothesis.
추천 영상:
가이드 코스
06:21
Step 1: Write Hypotheses

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. Many statistical tests, including those for comparing means, assume that the populations being studied are normally distributed. This assumption is important for the validity of the results when analyzing the differences in migraine days before and after treatment.
추천 영상:
가이드 코스
03:17
Finding Z-Scores for Non-Standard Normal Variables
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교과서 질문

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교과서 질문

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Assume the samples are random and independent, and the populations are normally distributed.

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교과서 질문

Testing the Difference Between Two Means (e) decide whether to reject or fail to reject the null hypothesis. Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)


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