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)


Better Tips by Giving Candy An experiment was conducted to determine whether giving candy to dining parties resulted in greater tips. The mean tip percentages and standard deviations are given below along with the sample sizes (based on data from “Sweetening the Till: The Use of Candy to Increase Restaurant Tipping,” by Strohmetz et al., Journal of Applied Social Psychology, Vol. 32, No. 2).


a. Use a 0.05 significance level to test the claim that giving candy does result in greater tips.

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Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ₁ = μ₂ (mean tip percentages are the same for no candy and two candies). The alternative hypothesis is H₁: μ₁ < μ₂ (mean tip percentage is greater for two candies).
2. Step 2: Determine the test statistic formula for comparing two independent sample means when population standard deviations are not assumed to be equal. Use the formula: t = (x₁ - x₂) / sqrt((s₁²/n₁) + (s₂²/n₂)), where x₁ and x₂ are sample means, s₁ and s₂ are sample standard deviations, and n₁ and n₂ are sample sizes.
3. Step 3: Calculate the degrees of freedom (df) using the formula: df = [(s₁²/n₁ + s₂²/n₂)²] / {[(s₁²/n₁)² / (n₁ - 1)] + [(s₂²/n₂)² / (n₂ - 1)]}. This will help determine the critical t-value from the t-distribution table.
4. Step 4: Use the significance level (α = 0.05) and the calculated degrees of freedom to find the critical t-value from the t-distribution table. Compare the calculated t-statistic to the critical t-value to determine whether to reject or fail to reject the null hypothesis.
5. Step 5: Interpret the results. If the calculated t-statistic exceeds the critical t-value, reject the null hypothesis and conclude that giving candy results in greater tips. Otherwise, fail to reject the null hypothesis and conclude that there is insufficient evidence to support the claim.

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 dining parties receiving candy and those not receiving candy are treated as separate groups, allowing for the comparison of their mean tip percentages without any influence from one another.

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 would state that giving candy does not affect tip percentages, while the alternative hypothesis posits that it does, guiding the analysis of the collected data.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether the results of a hypothesis test are statistically significant. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none, which is the criterion used to evaluate the results of the tipping experiment.