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.


c. Does the background color appear to have an effect on word recall scores? If so, which color appears to be associated with higher word memory recall scores?

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that the background color has no effect on word recall scores (mean scores are equal for red and blue backgrounds). The alternative hypothesis (H₁) states that the background color does have an effect (mean scores are not equal).
2. Step 2: Use the given data to calculate the test statistic. The formula for the test statistic is: $\frac{(x_1-x_2)}{\sqrt{\frac{s_1²/n_1+s_2²/n_2}{1}}}$, where x̄₁ and x̄₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes.
3. Step 3: Determine the degrees of freedom (df) using the formula: $\mathrm{df}=\frac{(s_1²/n_1+s_2²/n_2)²}{(\frac{(s_1²/n_1)²}{n_1-1}+\frac{(s_2²/n_2)²}{n_2-1})}$. Round df to the nearest integer.
4. Step 4: Use the calculated test statistic and degrees of freedom to find the p-value. This can be done using statistical software or a t-distribution table. Compare the p-value to the significance level (typically α = 0.05) to determine whether to reject the null hypothesis.
5. Step 5: Interpret the results. If the null hypothesis is rejected, conclude that the background color has an effect on word recall scores. Compare the sample means (x̄₁ = 15.89 for red and x̄₂ = 12.31 for blue) to determine which color is associated with higher word recall scores.

Key Concepts

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In this context, the red and blue background groups are independent, meaning the performance of one group does not influence the other. This is crucial for statistical tests that compare means, as it allows for the assumption that the samples can be treated separately.

Mean and Standard Deviation

The mean (average) is a measure of central tendency that summarizes the data by providing a single value representing the center of the dataset. The standard deviation measures the amount of variation or dispersion in a set of values. In this study, the means and standard deviations for word recall scores under red and blue backgrounds help assess the effect of color on cognitive performance.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this scenario, the null hypothesis might state that background color has no effect on word recall scores, while the alternative hypothesis suggests that it does. Analyzing the means and conducting a statistical test will help answer whether the background color significantly affects recall.