Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problema 9.4.17a

Capitolo 9, Problema 9.4.17a

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).
a. For each value x in the first sample, find the absolute deviation |x-x_bar| then sort the absolute deviation values. Do the same for the second sample.

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Calculate the sample mean \( \bar{x} \) for the first sample using the formula: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \] where \( n \) is the number of observations in the first sample and \( x_i \) are the individual observations.

For each observation \( x_i \) in the first sample, compute the absolute deviation from the mean: \[ |x_i - \bar{x}| \] This measures how far each data point is from the sample mean regardless of direction.

Repeat steps 1 and 2 for the second sample: calculate its sample mean \( \bar{y} \) and then find the absolute deviations \( |y_i - \bar{y}| \) for each observation \( y_i \) in the second sample.

Sort the absolute deviations for both samples in ascending order. This will help identify the largest deviations in each sample.

Count how many of the largest absolute deviations (usually the top 5) come from each sample. The 'count five' test compares these counts to assess if the variation between the two populations is significantly different.

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Absolute Deviation

Absolute deviation measures how far each data point is from the sample mean, ignoring direction. It is calculated as the absolute value of the difference between each observation and the mean, providing a way to assess variability without squaring differences as in variance.

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08:45

Calculating Standard Deviation

Comparing Variability Between Two Samples

Comparing variability involves assessing whether two populations have similar spread or dispersion. Traditional methods include the F test, but alternative approaches like the Count Five Test use order statistics of absolute deviations to test equality of variances without relying on normality assumptions.

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03:50

Probabilities Between Two Values

Count Five Test Procedure

The Count Five Test is a nonparametric method to compare variances by counting how many of the largest absolute deviations come from each sample. If one sample contributes more than five of the largest deviations, it suggests unequal variability, providing a simple alternative to the F test.

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04:04

Fundamental Counting Principle

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Pulse Rates of Women and Men Using the samples of women and men included in Data Set 1 “Body Data,” we get this 95% confidence interval estimate of the difference between the population mean of pulse rates (bpm) of women and the population mean of pulse rates (bpm) of men: 1.7 bpm < u1-u2 < 7.2bpm. In this confidence interval, women correspond to population 1 and men correspond to population 2.

a. What does the confidence interval suggest about equality of the mean pulse rate of women and the mean pulse rate of men?

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Friday the 13th Refer to the sample data from Exercise 1.

a. Find the differences d, then find the values of d_bar and sd

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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)

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.

a. Use a 0.05 significance level to test the claim that the samples are from populations with the same mean.

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Are Seat Belts Effective? A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed (based on data from “Who Wants Airbags?” by Meyer and Finney, Chance, Vol. 18, No. 2). We want to use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities.

a. Test the claim using a hypothesis test.

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Independent Samples Which of the following involve independent samples?

a. Data Set 4 “Measured and Reported” includes measured heights matched with the heights that were reported when the subjects were asked for those values.

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Hypotheses and Conclusions Refer to the hypothesis test described in Exercise 1.

a. Identify the null hypothesis and the alternative hypothesis.

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