Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 9.2.5b

Chapter 9, Problem 9.2.5b

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).

b. Construct the confidence interval suitable for testing the claim in part (a).

Verified step by step guidance

Step 1: Identify the given data for both groups. For the 'No Candy' group: sample size (n₁) = 20, mean (x̄₁) = 18.95, standard deviation (s₁) = 1.50. For the 'Two Candies' group: sample size (n₂) = 20, mean (x̄₂) = 21.62, standard deviation (s₂) = 2.51.

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 table based on degrees of freedom (df).

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 determine the appropriate t-value for the confidence interval.

Step 4: Find the critical t-value corresponding to the desired confidence level (e.g., 95%) and the calculated degrees of freedom. Use a t-distribution table or technology to find this value.

Step 5: Substitute the values for x̄₁, x̄₂, s₁, s₂, n₁, n₂, and the critical t-value into the confidence interval formula to compute the interval. Interpret the interval in the context of the problem to determine whether giving candy results in greater tips.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is calculated using the sample mean, standard deviation, and sample size, along with a critical value from the t-distribution or z-distribution, depending on the sample size and whether the population standard deviation is known. In this context, constructing a confidence interval will help assess the difference in tip percentages between the two groups.

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In this experiment, the tips from dining parties that received candy are independent of those that did not. This independence is crucial for applying statistical tests, as it ensures that the results from one sample do not influence the results from another, allowing for valid comparisons between the groups.

T-test for Independent Samples

A t-test for independent samples is a statistical method used to determine if there is a significant difference between the means of two independent groups. It is particularly useful when the population standard deviations are unknown and the sample sizes are small. In this case, the t-test will help evaluate whether the difference in mean tip percentages between the groups (with and without candy) is statistically significant.

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b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

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b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

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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)

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b. Construct the confidence interval suitable for testing the claim in part (a).

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b. Construct the confidence interval suitable for testing the claim in part (a).

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b. Construct the confidence interval suitable for testing the claim in part (a).

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