Skip to main content
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.T.3e
Chapter 10, Problem 10.T.3e

In each exercise,
e. interpret the decision in the context of the original claim.
[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)
Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32
Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80
Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50
Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

Verified step by step guidance
1
Step 1: State the hypotheses for the ANOVA test. The null hypothesis \(H_0\) is that the mean hourly wages are the same for all three states: \(\mu_{Maine} = \mu_{Oklahoma} = \mu_{Massachusetts}\). The alternative hypothesis \(H_a\) is that at least one state's mean wage is different.
Step 2: Calculate the sample means and sample variances for each state using the given wage data. This involves summing the wages for each state and dividing by the number of observations to find the means, and then computing the variance for each sample.
Step 3: Compute the overall mean wage by combining all the data from the three states. This is done by summing all wages from all states and dividing by the total number of observations.
Step 4: Calculate the Between-Group Sum of Squares (SSB) and the Within-Group Sum of Squares (SSW). Use the formulas: \[SSB = \sum_{i=1}^k n_i (\bar{x}_i - \bar{x})^2\] \[SSW = \sum_{i=1}^k \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2\] where \(k\) is the number of groups (3 states), \(n_i\) is the sample size for group \(i\), \(\bar{x}_i\) is the sample mean for group \(i\), and \(\bar{x}\) is the overall mean.
Step 5: Calculate the Mean Squares for Between-Groups (MSB) and Within-Groups (MSW) by dividing the sums of squares by their respective degrees of freedom: \[MSB = \frac{SSB}{k-1}\] \[MSW = \frac{SSW}{N-k}\] where \(N\) is the total number of observations. Then compute the F-statistic: \[F = \frac{MSB}{MSW}\] Compare this F-statistic to the critical value from the F-distribution with \(k-1\) and \(N-k\) degrees of freedom at \(\alpha=0.01\). If \(F\) is greater than the critical value, reject \(H_0\); otherwise, do not reject \(H_0\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

One-Way ANOVA (Analysis of Variance)

One-Way ANOVA is a statistical method used to compare the means of three or more independent groups to determine if at least one group mean differs significantly. It tests the null hypothesis that all group means are equal by analyzing variance within and between groups. This method is appropriate when samples are independent and populations are normally distributed with equal variances.
Recommended video:
06:46
Introduction to ANOVA

Assumption of Equal Population Variances

The assumption of equal variances (homogeneity of variance) means that the variability within each group is similar across all groups. This is crucial for the validity of ANOVA results because unequal variances can affect the test’s accuracy. Tests like Levene’s test can check this assumption before performing ANOVA.
Recommended video:
04:48
Population Standard Deviation Known

Significance Level (α) and Hypothesis Testing

The significance level, α, is the threshold for deciding whether to reject the null hypothesis. Here, α=0.01 means there is a 1% risk of concluding that the means differ when they actually do not. If the ANOVA p-value is less than α, we reject the null hypothesis and interpret the decision in the context of the original claim about wage equality.
Recommended video:
05:52
Performing Hypothesis Tests: Proportions
Related Practice
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (b) find the critical value and identify the rejection region.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


49
views
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


71
views
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=20,d.f.D=25

78
views
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (c) find the chi-square test statistic.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


67
views
Textbook Question

In each exercise,

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

[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

21
views
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=0.01,d.f.N=40,d.f.D=60

63
views