Question

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)

Color and Cognition Researchers from the University of British Columbia conducted a study to investigate the effects of color on cognitive tasks. Words were displayed on a computer screen with background colors of red and blue. Results from scores on a test of word recall are given below. Higher scores correspond to greater word recall.

a. Use a 0.05 significance level to test the claim that the samples are from populations with the same mean.

Accepted Answer

Step 1: State the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ₁ = μ₂ (the population means are equal), and the alternative hypothesis is H₁: μ₁ ≠ μ₂ (the population means are not equal). Step 2: Identify the significance level (α). The problem specifies a significance level of 0.05. Step 3: Calculate the test statistic using the formula for a two-sample t-test for independent samples: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)), where x̄₁ and x̄₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes. Step 4: Determine the degrees of freedom (df) using the formula: df = min(n₁ - 1, n₂ - 1). In this case, df = min(35 - 1, 36 - 1) = 34. Step 5: Compare the calculated t-value to the critical t-value from the t-distribution table at df = 34 and α = 0.05 (two-tailed test). If the calculated t-value exceeds the critical t-value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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Key Concepts

Independent Samples

Hypothesis Testing

t-Test for Independent Samples