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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 10.4.18
Chapter 10, Problem 10.4.18

"Using Technology to Perform a Two-Way ANOVA Test In Exercises 15–18, use technology and the block design to perform a two-way ANOVA test. Use α=0.10. Interpret the results. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.


[APPLET] Laptop Repairs The manager of a computer repair service wants to determine whether there is a difference in the time it takes four technicians to repair different brands of laptops. The block design shows the times (in minutes) it took for each technician to repair three laptops of each brand.

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Step 1: Organize the data into a two-way ANOVA format. The rows represent the laptop brands (Brand A, Brand B, Brand C), and the columns represent the technicians (Technician 1, Technician 2, Technician 3, Technician 4). Each cell contains the repair times for the respective brand and technician.
Step 2: Calculate the mean repair time for each brand (row means) and each technician (column means). This helps to understand the overall trends in the data.
Step 3: Use technology (e.g., statistical software or a calculator) to compute the sum of squares for the factors (brands and technicians), the interaction between the factors, and the error term. These values are necessary for the ANOVA table.
Step 4: Construct the ANOVA table, including the degrees of freedom, sum of squares, mean squares, F-statistic, and p-value for each factor (brands and technicians) and their interaction. Use α=0.10 as the significance level.
Step 5: Interpret the results. Compare the p-values for the factors and their interaction to α=0.10. If the p-value is less than α, conclude that there is a significant difference for that factor or interaction. Otherwise, conclude that there is no significant difference.

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Key Concepts

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

Two-Way ANOVA

Two-Way ANOVA (Analysis of Variance) is a statistical method used to determine the effect of two independent categorical variables on a continuous dependent variable. It helps in assessing whether there are any statistically significant differences between the means of different groups. In this case, the two factors are the technicians and the brands of laptops, and the dependent variable is the time taken for repairs.
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Block Design

Block design is an experimental design technique that involves grouping similar experimental units into blocks to reduce variability. In this scenario, each technician represents a block, and the aim is to control for the variability in repair times that may arise from differences among technicians. This design allows for a clearer analysis of the effect of the brands on repair times while accounting for technician differences.
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Critical Values: t-Distribution

Assumptions of ANOVA

ANOVA relies on several key assumptions: the samples must be independent, the populations should be normally distributed, and the population variances should be equal (homogeneity of variance). In this exercise, these assumptions are crucial for the validity of the test results. If these assumptions are violated, the results of the ANOVA may not be reliable, leading to incorrect conclusions about the differences in repair times.
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Related Practice
Textbook Question

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

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)

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 = 0.05.

Sample statistics: s₁² = 245, n₁ = 31 and s₂² = 112, n₂ = 28

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