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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
All textbooksTriola 14th EditionCh. 9 - Inferences from Two SamplesProblem 9.4.1d
Chapter 9, Problem 9.4.1d

F Test Statistic


d. Is the F distribution symmetric, skewed left, or skewed right?

Verified step by step guidance
1
Understand the F distribution: The F distribution is used in statistics to compare variances between two groups or to test hypotheses in analysis of variance (ANOVA). It is defined by two degrees of freedom parameters, one for the numerator and one for the denominator.
Recognize the shape of the F distribution: The F distribution is not symmetric. It is skewed to the right, meaning it has a longer tail on the right side of the distribution.
Consider why the F distribution is skewed: The skewness arises because the F statistic is a ratio of two independent chi-squared distributions divided by their respective degrees of freedom. Ratios of squared values tend to produce a distribution with a right skew.
Note the range of the F distribution: The F statistic can only take positive values (greater than or equal to 0), as variances and squared values cannot be negative. This contributes to the right skew of the distribution.
Apply this knowledge in practice: When interpreting results involving the F distribution, remember its skewed nature and the fact that larger F values indicate greater evidence against the null hypothesis in hypothesis testing scenarios.

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Key Concepts

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

F Distribution

The F distribution is a probability distribution that arises frequently in statistics, particularly in the context of variance analysis. It is defined by two parameters, the degrees of freedom for the numerator and the denominator, which correspond to the variances being compared. The F distribution is used to determine if there are significant differences between the variances of two or more groups.
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Skewness

Skewness refers to the asymmetry of a probability distribution. A distribution is said to be skewed left (negatively skewed) if it has a longer tail on the left side, while it is skewed right (positively skewed) if the tail is longer on the right side. Understanding skewness is crucial for interpreting the shape of the distribution and its implications for statistical analysis.
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Properties of the F Distribution

The F distribution is inherently right-skewed, meaning it has a longer tail on the right side. This characteristic is important when conducting hypothesis tests, as it affects the critical values used to determine statistical significance. The right skewness indicates that most of the values are concentrated on the left, with fewer extreme values on the right, which is typical in variance comparisons.
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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).

d. If c1 equal to or greater than critical value then conclude that sigma2,1 > sigma2,2 If c1 equal to or greater than critical value then conclude that sigma2,2 > sigma2,1. Otherwise, fail to reject the null hypothesis of sigma2,1 = sigma2,2

Textbook Question

P-VALUE The test statistic of z = 2.14 is obtained when using the data from Exercise 1 and testing the claim that patients treated with dexamethasone and patients given a placebo have the same rate of complete resolution.


a. Find the P-value for the test.

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).

c. If the sample sizes are equal (n1 = n2) use a critical value of 5. If n1 is not equals to n2 calculate the critical value shown below.

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)


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.


c. Does the background color appear to have an effect on word recall scores? If so, which color appears to be associated with higher word memory recall scores?


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Textbook Question

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.


a. Test the claim using a hypothesis test.

Textbook Question

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.


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