In Exercises 5–20, assume that the two samples are independent...

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)

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.

Comparison of tip percentages: No Candy (n=20, x̄=18.95, s=1.50) vs. Two Candies (n=20, x̄=21.62, s=2.51).

[Image]

Accepted Answer

All right, hello, everyone. So this question says, an experiment was conducted to determine whether providing candy to dining parties resulted in a higher tip percentage. The following data were collected on the tip percentages for two independent samples of dining parties. Use a 0.05 significance level to test the claim that giving candy results in greater tips. So first, what are the possible hypotheses, right? First, we have the null hypothesis or H not. Which would state That mu 1 is equal to mu 2. Which means that the tip percentage or the mean tip percentage with and without candy is the same. Now, the alternative hypothesis on the other hand, would state that mu 1 is less than m2. So the alternative hypothesis is saying that the mean tip percentage with candy is greater than the one without. And so because we're testing, if the mean tip percentage increases with candy, this is a right tailed test. So let's begin with the test statistic. So, recall that because we're dealing with two independent samples, we're going to use the following formula for our T statistic. So tea is equal to X bar 1 subtracted by X 2. Divided by The square root of S 1 squared. Divided by N1, added to S2 squad, divided by N2. So now let's plug in the information that we already have right here. Sample 1 corresponds to no candy and sample 2 is where candy was given. So X bar 1, which is the mean tip percentage, is 15. Subtracted by that of sample 2, which is 18. So now we divide this by. The sum of The standard deviations of both samples squared, divided by their respective sample size. So that's the square root of in the denominator. Four squared Divided by 40 Added to 5 squared Divided by 45. And so evaluating this expression equals approximately -3.07. Which means that we now have to compare our T value of -3.07 to the critical value. But here's the thing, right? The fact that we're not Assuming that the population distributions. Or rather, because we're assuming. That both population standard deviations are not equal, this changes the formula for degrees of freedom. So the degrees of freedom are equal to. Let's see, S1 squared divided by N1. Added to S2 2 divided by N2. Squared Divided by S1 squared divided by N1. Squared Divided by And one subtracted by one. And added to S 2 squad Divided by N2. Squared Divided by And 2 subtracted by 1. So substituting the given values and evaluating this expression gives you approximately 82 degrees of freedom. Which now leads you to find the critical values. So, we can actually use the T distribution table to find the critical values for a right-tailed test with alpha equals 0.05 and 82 degrees of freedom. So, that critical value. Is equal to approximately 1.664. Which now leads you into your decision. Right. Now, the test statistic that we calculated earlier has to be greater than the critical value for us to be able to reject the null hypothesis. But in this case, our test value of -3.07 is less than our critical value, which means that we failed to reject the null hypothesis. So now this leads us to our final answer, which reads as follows. We fail to reject H knot. The candy does not result in greater tips. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Independent Samples

Hypothesis Testing

Significance Level

Watch next