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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 9 - Inferences from Two SamplesProblem 9.4.7
Chapter 9, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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
Related Practice
Textbook Question

Equivalence of Hypothesis Test and Confidence Interval Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1=p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1-p2

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Textbook Question

Bootstrapping and Randomization When resampling data from two independent samples, what is the fundamental difference between bootstrapping and randomization?

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Textbook Question

In Exercises 5–8, use (a) randomization and (b) bootstrapping for the indicated exercise from Section 9-1. Compare the results to those obtained in the original exercise.


Exercise 7 in Section 9-1 “Buttered Toast Drop”

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Textbook Question

Test for Normality For the hypothesis test described in Exercise 2, the sample sizes are n1 = 2208 and n2 = 1986 When using the F test with these data, is it correct to reason that there is no need to check for normality because both samples have sizes that are greater than 30?

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Textbook Question

Randomization vs t Test Two samples of commute times from Boston and New York are randomly selected and it is found that the samples sizes are n1 = 18 and n2 = 12 and each of the two samples appears to be from a population with a distribution that is dramatically far from normal. Which method is more likely to yield better results for testing Mu1 is not equals to Mu2. Hypothesis test using the t distribution (as in Section 9-2) or the resampling method?

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Textbook Question

Robust What does it mean when we say that the F test described in this section is not robust against departures from normality?

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