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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Triola 14th EditionCh. 9 - Inferences from Two Samples문제 9.4.7
9장, 문제 9.4.7

Color and Creativity Researchers from the University of British Columbia conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given below. Use a 0.05 significance level to test the claim that creative task scores have the same variation with a red background and a blue background.
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Step 1: Identify the hypothesis test to be used. Since the problem involves comparing the variation (or variance) of two groups (red background and blue background), we will use an F-test for equality of variances.
Step 2: State the null and alternative hypotheses. The null hypothesis (H₀) is that the variances of the two groups are equal (σ₁² = σ₂²). The alternative hypothesis (H₁) is that the variances are not equal (σ₁² ≠ σ₂²).
Step 3: Calculate the test statistic. The F-test statistic is calculated as the ratio of the larger sample variance to the smaller sample variance: F = s₁² / s₂², where s₁² and s₂² are the sample variances of the two groups. Ensure you identify which group has the larger variance.
Step 4: Determine the critical value or p-value. Using the F-distribution table, find the critical value for the given degrees of freedom (df₁ = n₁ - 1 and df₂ = n₂ - 1, where n₁ and n₂ are the sample sizes of the two groups) and the significance level (α = 0.05). Alternatively, calculate the p-value using statistical software.
Step 5: Make a decision. Compare the test statistic to the critical value or compare the p-value to the significance level. If the test statistic exceeds the critical value or if the p-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Conclude whether there is evidence to suggest that the variances of the creative task scores differ between the red and blue backgrounds.

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주요 개념

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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. In this context, the null hypothesis would state that there is no difference in creativity scores between the red and blue backgrounds.
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Step 1: Write Hypotheses

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. A common significance level is 0.05, which implies that there is a 5% risk of concluding that a difference exists when there is none. In this study, using a 0.05 significance level means that if the p-value is less than 0.05, the researchers would reject the null hypothesis and conclude that the background color affects creativity scores.
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가이드 코스
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Step 4: State Conclusion Example 4

Variance and Comparison of Variances

Variance measures the spread of a set of data points around their mean, indicating how much the scores differ from each other. In this scenario, comparing the variances of creativity scores from the two different background colors is essential to determine if the variability in scores is significantly different. Statistical tests, such as Levene's test or the F-test, can be used to assess whether the variances are equal, which is a prerequisite for many parametric tests.
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Variance & Standard Deviation of Discrete Random Variables
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