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.


b. Construct a confidence interval appropriate for the hypothesis test in part (a). What is it about the confidence interval that causes us to reach the same conclusion from part (a)?

Verified step by step guidance

1. Step 1: Identify the given data for the two independent samples. For the red background: sample size (n₁) = 35, sample mean (x̄₁) = 15.89, and sample standard deviation (s₁) = 5.90. For the blue background: sample size (n₂) = 36, sample mean (x̄₂) = 12.31, and sample standard deviation (s₂) = 5.48.
2. Step 2: Use the formula for the confidence interval for the difference between two means when the population standard deviations are not assumed to be equal. The formula is: CI = (x̄₁ - x̄₂) ± t * √((s₁² / n₁) + (s₂² / n₂)), where t is the critical value from the t-distribution with degrees of freedom calculated using the Welch-Satterthwaite equation.
3. Step 3: Calculate the degrees of freedom (df) using the Welch-Satterthwaite equation: df = ((s₁² / n₁) + (s₂² / n₂))² / {[(s₁² / n₁)² / (n₁ - 1)] + [(s₂² / n₂)² / (n₂ - 1)]}. This value will determine the t critical value for the desired confidence level.
4. Step 4: Determine the t critical value for the desired confidence level (e.g., 95%) using the degrees of freedom calculated in Step 3 and a t-distribution table or statistical software.
5. Step 5: Substitute the values for x̄₁, x̄₂, s₁, s₂, n₁, n₂, and the t critical value into the confidence interval formula from Step 2. Simplify the expression to find the confidence interval. Compare the confidence interval to the null hypothesis value (e.g., 0) to confirm the conclusion from part (a).

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It provides an estimate of uncertainty around a sample mean, allowing researchers to infer about the population. The width of the interval reflects the level of confidence; a wider interval indicates more uncertainty, while a narrower interval suggests more precision.

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In statistical analysis, this means that the selection of one sample does not influence the selection of another. This concept is crucial when comparing means from different groups, as it allows for the application of specific statistical tests, such as the t-test, to determine if there are significant differences between the groups.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (no effect or difference) and an alternative hypothesis (some effect or difference), then using sample data to determine whether to reject the null hypothesis. The results of hypothesis tests are often supported by confidence intervals, which provide additional context for the findings.