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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
Chapter 9, Problem 9.2.18b

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)


Queues Listed on the next page are waiting times (seconds) of observed cars at a Delaware inspection station. The data from two waiting lines are real observations, and the data from the single waiting line are modeled from those real observations. These data are from Data Set 30 “Queues” in Appendix B. The data were collected by the author.


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


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Step 1: Organize the data into two groups: 'Two Lines' and 'One Line'. Calculate the sample size (n), mean (x̄), and standard deviation (s) for each group. Use the formulas for mean and standard deviation: Mean: x̄ = (Σx)/n, Standard Deviation: s = sqrt((Σ(x - x̄)^2)/(n-1)).
Step 2: Identify the degrees of freedom (df) for the t-distribution. Since the population standard deviations are not assumed to be equal, use the formula for approximate degrees of freedom: df = ((s1^2/n1 + s2^2/n2)^2) / (((s1^2/n1)^2/(n1-1)) + ((s2^2/n2)^2/(n2-1))).
Step 3: Determine the critical t-value for the desired confidence level (e.g., 95%) using the degrees of freedom calculated in Step 2. Refer to a t-distribution table or use statistical software to find the critical value.
Step 4: Calculate the margin of error (E) using the formula: E = t * sqrt((s1^2/n1) + (s2^2/n2)), where t is the critical t-value, s1 and s2 are the standard deviations of the two groups, and n1 and n2 are their respective sample sizes.
Step 5: Construct the confidence interval for the difference in means (μ1 - μ2) using the formula: (x̄1 - x̄2) ± E, where x̄1 and x̄2 are the sample means of the two groups, and E is the margin of error calculated in Step 4.

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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 constructed using the sample mean, the standard error, and a critical value from the t-distribution or z-distribution, depending on the sample size and whether the population standard deviation is known. The width of the interval reflects the level of confidence; a higher confidence level results in a wider interval.
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Independent Samples

Independent samples refer to two or more groups of data that are collected separately and do not influence each other. In statistical analysis, this assumption is crucial when comparing means or proportions, as it allows for the application of specific tests, such as the t-test for independent samples. The independence of samples ensures that the results are not biased by any relationship between the groups.
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Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean and the standard deviation. Many statistical methods, including confidence intervals and hypothesis tests, assume that the underlying data are normally distributed, especially when sample sizes are small.
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Related Practice
Textbook Question

Count Five Test for Comparing Variation in Two Populations Repeat Exercise 16 “Blanking Out on Tests,” but instead of using the F test, use the following procedure for the “count five” test of equal variations (which is not as complicated as it might appear).

b. Let c1 be the count of the number of absolute deviation values in the first sample that are greater than the largest absolute deviation value in the other sample. Also, let C2 be the count of the number of absolute deviation values in the second sample that are greater than the largest absolute deviation value in the other sample. (One of these counts will always be zero.)

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Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.


Measured and Reported Weights Listed below are measured and reported weights (lb) of random female subjects (from Data Set 4 “Measured and Reported” in Appendix B).


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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Textbook 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)


Magnet Treatment of Pain People spend around \$5 billion annually for the purchase of magnets used to treat a wide variety of pains. Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Higher scores correspond to greater pain levels.


b. Construct the confidence interval appropriate for the hypothesis test in part (a).


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Textbook 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).


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


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Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.


Do Men Talk Less than Women? Listed below are word counts of males and females in couple relationships (from Data Set 14 “Word Counts” in Appendix B).


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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Textbook 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)


Readability of Font On a Computer Screen The statistics shown below were obtained from a standard test of readability of fonts on a computer screen (based on data from “Reading on the Computer Screen: Does Font Type Have Effects on Web Text Readability?” by Ali et al., International Education Studies, Vol. 6, No. 3). Reading speed and accuracy were combined into a readability performance score (x), where a higher score represents better font readability.


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


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