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Ch. 10 - Chi-Square Tests and the F-Distribution
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 10 - Chi-Square Tests and the F-DistributionProblem 10.R.11
Chapter 10, Problem 10.R.11

"In Exercises 9–12, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.10,d.f.N=5,d.f.D=12"

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Step 1: Understand the problem. We are tasked with finding the critical F-value for a right-tailed test given the level of significance (α = 0.10), degrees of freedom for the numerator (d.f.N = 5), and degrees of freedom for the denominator (d.f.D = 12).
Step 2: Recall the definition of the F-distribution. The F-distribution is used in hypothesis testing, particularly in ANOVA, and depends on two sets of degrees of freedom: one for the numerator (d.f.N) and one for the denominator (d.f.D).
Step 3: Use the F-distribution table or statistical software. To find the critical F-value, locate the row corresponding to d.f.N = 5 and the column corresponding to d.f.D = 12 in the F-distribution table for α = 0.10 (right-tailed test).
Step 4: If using statistical software (e.g., Excel, R, or a calculator), use the appropriate function. For example, in Excel, you can use the formula F.INV.RT(0.10, 5, 12) to find the critical F-value.
Step 5: Interpret the result. The critical F-value represents the threshold beyond which the test statistic would lead to rejecting the null hypothesis in a right-tailed test at the 0.10 significance level.

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Key Concepts

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

Critical F-value

The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis in an F-test. It is derived from the F-distribution, which is used to compare variances between two groups. The critical value is dependent on the chosen significance level (α) and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D). If the calculated F-statistic exceeds this critical value, the null hypothesis is rejected.
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Degrees of Freedom

Degrees of freedom (d.f.) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of an F-test, d.f.N represents the degrees of freedom associated with the numerator (the group with more variability), while d.f.D represents the degrees of freedom for the denominator (the group with less variability). These values are crucial for determining the shape of the F-distribution and finding the critical F-value.
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Significance Level (α)

The significance level (α) is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. It is a threshold set by the researcher, commonly at values like 0.05 or 0.10, which indicates the level of risk the researcher is willing to take. In this case, with α set at 0.10, it means there is a 10% chance of incorrectly rejecting the null hypothesis, guiding the decision-making process in hypothesis testing.
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Related Practice
Textbook Question

In Exercises 21 and 22, (d) decide whether to reject or fail to reject the null hypothesis,

Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)

20
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Textbook Question

In Exercises 17–20, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.


[APPLET] An instructor claims that the variance of SAT evidence-based reading and writing scores is different than the variance of SAT math scores. The table shows the SAT evidence-based reading and writing scores for 12 randomly selected students and the SAT math scores for 12 randomly selected students. At α=0.01, can you support the instructor’s claim?


78
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Textbook Question

In Exercises 21 and 22, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.


[APPLET] The table shows the annual incomes (in dollars) for a sample of families from four regions of the United States. At α=0.05, can you conclude that the mean annual income of families is different in at least one of the regions? (Adapted from U.S. Census Bureau)


53
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Textbook Question

"In Exercises 13–16, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.10,d.f.N=15,d.f.D=27"

86
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Textbook Question

In Exercises 21 and 22, (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)

30
views
Textbook Question

In Exercises 21 and 22, (c) find the test statistic F, Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)

39
views