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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 10.1.6
Chapter 10, Problem 10.1.6

Finding Expected Frequencies
In Exercises 3–6, find the expected frequency for the values of n and pᵢ.


n=415, pᵢ=0.08

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1
Identify the formula for calculating the expected frequency: Expected Frequency (E) = n × pᵢ, where n is the total number of observations and pᵢ is the probability of the specific category.
Substitute the given values into the formula. Here, n = 415 and pᵢ = 0.08.
Perform the multiplication: Multiply 415 by 0.08 to calculate the expected frequency.
Interpret the result: The expected frequency represents the number of occurrences you would expect in the specific category based on the given probability and total observations.
Ensure the result is reasonable: Verify that the expected frequency is a positive value and makes sense in the context of the problem.

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Key Concepts

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

Expected Frequency

Expected frequency is a statistical term that refers to the anticipated number of occurrences of an event in a given sample size, based on a specific probability. It is calculated by multiplying the total number of observations (n) by the probability of the event (pᵢ). In this case, with n=415 and pᵢ=0.08, the expected frequency can be found by calculating 415 * 0.08.
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Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. A probability of 0 indicates that the event will not occur, while a probability of 1 indicates certainty. In the context of this question, pᵢ=0.08 signifies that there is an 8% chance of the event happening in any given trial.
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Sample Size

Sample size refers to the number of observations or data points collected in a study or experiment. It is crucial for determining the reliability and validity of statistical results. In this question, n=415 indicates that the analysis is based on 415 observations, which influences the calculation of expected frequencies and the overall statistical inference.
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Related Practice
Textbook Question

"In Exercises 13–18, test the claim about the difference between two population variances σ₁² and σ₂² at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: σ₁² > σ₂²; α = 0.10.

Sample statistics: s₁² = 773, n₁ = 5 and s₂² = 765, n₂ = 6"

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Textbook Question

Conditional Relative Frequencies In Exercises 37–42, use the contingency table from Exercises 33–36, and the information below.

Relative frequencies can also be calculated based on the row totals (by dividing each row entry by the row’s total) or the column totals (by dividing each column entry by the column’s total). These frequencies are conditional relative frequencies and can be used to determine whether an association exists between two categories in a contingency table.


What percent of U.S. adults ages 25 and over who have a degree are not in the civilian labor force?

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Textbook Question

Finding Expected Frequencies

In Exercises 3–6, find the expected frequency for the values of n and pᵢ.


n=230, pᵢ=0.25

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Textbook Question

"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, 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=24, d.f.D=28"

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Textbook Question

State the null and alternative hypotheses for a one-way ANOVA test.

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Textbook Question

Performing a Chi-Square Independence Test In Exercises 13–28, perform the indicated chi-square independence test by performing the steps below.

a. Identify the claim and state H₀ and Hₐ


b. Determine the degrees of freedom, find the critical value, and identify the rejection region.


c. Find the chi-square test statistic.


d. Decide whether to reject or fail to reject the null hypothesis.


e. Interpret the decision in the context of the original claim.


Alcohol-Related Accidents The contingency table shows the results of a random sample of fatally injured passenger vehicle drivers (with blood alcohol concentrations greater than or equal to 0.08) by age and gender. At α=0.05, can you conclude that age is related to gender in such alcohol-related accidents? (Adapted from Insurance Institute for Highway Safety)

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