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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 9 - Inferences from Two SamplesProblema 9.4.1b
Capitolo 9, Problema 9.4.1b

F Test Statistic


b. Can the F test statistic ever be a negative number?

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Understand the F test statistic: The F test statistic is used in hypothesis testing to compare variances or to test the overall significance of a regression model. It is calculated as the ratio of two variances (or mean square values).
Recall the formula for the F test statistic: \( F = \frac{\text{Variance 1}}{\text{Variance 2}} \), where both variances are non-negative values because variances are squared quantities.
Recognize that since variances cannot be negative, the numerator and denominator in the F test statistic formula are always non-negative. This ensures that the F test statistic itself is always a non-negative value.
Understand the interpretation: A higher F value indicates a larger difference between the variances being compared, while an F value close to 1 suggests similar variances. However, the F test statistic cannot be negative because it is derived from non-negative quantities.
Conclude: The F test statistic can never be a negative number due to the mathematical properties of variances and the formula used to compute it.

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F Test Statistic

The F test statistic is a ratio used in statistical analysis to compare variances between two or more groups. It is calculated by dividing the variance between the group means by the variance within the groups. This statistic is commonly used in ANOVA (Analysis of Variance) to determine if there are significant differences among group means.
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Step 2: Calculate Test Statistic

Variance

Variance is a measure of the dispersion or spread of a set of data points in a dataset. It quantifies how much the values in a dataset differ from the mean of that dataset. In the context of the F test, variance is crucial as it helps assess the degree of variability within and between groups, which is essential for determining the significance of the F statistic.
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Variance & Standard Deviation of Discrete Random Variables

Non-negative Values

The F test statistic is always non-negative because it is derived from the ratio of variances, which are themselves always non-negative values. Since variances cannot be negative, the F statistic, being a ratio of these variances, cannot be negative either. This property is fundamental to the interpretation of the F test in hypothesis testing.
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Permutations of Non-Distinct Objects
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Domanda del libro di testo

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 Creativity Researchers from the University of British Columbia conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given below. Higher scores correspond to more creativity. The researchers make the claim that “blue enhances performance on a creative task.”


b. Construct the confidence interval appropriate for the hypothesis test in part (a). What is it about the confidence interval that causes us to reach the same conclusion from part (a)?


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Domanda del libro di testo

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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Domanda del libro di testo

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


b. In general, what does ud represent?

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Domanda del libro di testo

Cigarette Pack Warnings A study was conducted to find the effects of cigarette pack warnings that consisted of text or pictures. Among 1078 smokers given cigarette packs with text warnings, 366 tried to quit smoking. Among 1071 smokers given cigarette packs with warning pictures, 428 tried to quit smoking. (Results are based on data from “Effect of Pictorial Cigarette Pack Warnings on Changes in Smoking Behavior,” by Brewer et al., Journal of the American Medical Association.) Use a 0.01 significance level to test the claim that the proportion of smokers who tried to quit in the text warning group is less than the proportion in the picture warning group.


b. Test the claim by constructing an appropriate confidence interval.

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Domanda del libro di testo

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


b. If the P-value for the test is reported as “less than 0.001,” what should we conclude about the original claim?

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Domanda del libro di testo

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.


b. Test the claim by constructing an appropriate confidence interval.


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