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

Larson 8th Edition

Ch. 10 - Chi-Square Tests and the F-Distribution

Problem 10.Q.2a

Chapter 10, Problem 10.Q.2a

In each exercise,
a. identify the claim and state H₀ and Hₐ,

In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)

Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

Verified step by step guidance

Step 1: Identify the claim. The claim is that age and educational attainment are related. This means we are testing whether there is an association between the two variables.

Step 2: State the null hypothesis (H₀) and the alternative hypothesis (Hₐ). H₀: Age and educational attainment are not related (they are independent). Hₐ: Age and educational attainment are related (they are not independent).

Step 3: Organize the data into a contingency table. Use the provided table to structure the observed frequencies for the two age groups (30–34 and 65–69) across the educational attainment categories.

Step 4: Calculate the expected frequencies for each cell in the contingency table. Use the formula: Expected frequency = (Row total × Column total) / Grand total. This step ensures we have the expected values under the assumption of independence.

Step 5: Perform a chi-square test for independence. Compute the test statistic using the formula: χ² = Σ((Observed - Expected)² / Expected). Compare the calculated χ² value to the critical value from the chi-square distribution table at α = 0.01 with the appropriate degrees of freedom (df = (number of rows - 1) × (number of columns - 1)). If χ² > critical value, reject H₀; otherwise, fail to reject H₀.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H₀), which states there is no effect or relationship, and the alternative hypothesis (Hₐ), which suggests there is an effect or relationship. In this context, the goal is to determine if there is a significant relationship between age and educational attainment.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this case, α is set at 0.01, indicating a 1% risk of concluding that a relationship exists when there is none, thus requiring strong evidence to support the alternative hypothesis.

Contingency Table

A contingency table is a data representation that displays the frequency distribution of variables, allowing for the analysis of the relationship between them. In this exercise, the table compares educational attainment across different age groups, facilitating the examination of whether age influences educational achievement. This format is essential for conducting chi-square tests or other statistical analyses to assess associations.

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Contingency Tables & Expected Frequencies

Related Practice

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=0.05,d.f.N=9,d.f.D=8"

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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)

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Textbook Question

In each exercise,

c. find the test statistic,

In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)

Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

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Textbook Question

In each exercise,

d. decide whether to reject or fail to reject the null hypothesis, and

[APPLET] In Exercises 3 and 4, use the data, which list the annual wages (in thousands of dollars) for randomly selected individuals from three metropolitan areas. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Economic Analysis)

Ithaca, NY: 53.0, 60.3, 34.6, 37.1, 46.6, 46.8, 41.4, 50.6, 50.8, 49.4, 35.0, 36.7, 57.1

Little Rock, AR: 50.7, 43.7, 53.4, 40.0, 45.2, 52.7, 35.2, 60.4, 40.0, 45.9, 45.7, 47.3, 46.5, 44.5, 31.5

Madison, WI: 62.4, 53.9, 67.6, 52.9, 67.7, 50.7, 62.1, 58.9, 61.1, 65.0, 60.4, 59.6, 51.3, 44.8, 66.2

Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.

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Textbook Question

"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=0.05,d.f.N=6,d.f.D=50"

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Textbook Question

In each exercise,

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

Ithaca, NY: 53.0, 60.3, 34.6, 37.1, 46.6, 46.8, 41.4, 50.6, 50.8, 49.4, 35.0, 36.7, 57.1

Little Rock, AR: 50.7, 43.7, 53.4, 40.0, 45.2, 52.7, 35.2, 60.4, 40.0, 45.9, 45.7, 47.3, 46.5, 44.5, 31.5

Madison, WI: 62.4, 53.9, 67.6, 52.9, 67.7, 50.7, 62.1, 58.9, 61.1, 65.0, 60.4, 59.6, 51.3, 44.8, 66.2

Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.

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