In each exercise,
c. find the test statistic,
[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.

Verified step by step guidance

1. Step 1: Identify the problem type. Since we want to compare the mean annual wages of three independent groups (Ithaca, Little Rock, and Madison) and assume normality and equal variances, this calls for a one-way ANOVA test.
2. Step 2: Calculate the sample means (\(\bar{X}_1\), \(\bar{X}_2\), \(\bar{X}_3\)) and sample sizes (\(n_1\), \(n_2\), \(n_3\)) for each city using the given wage data.
3. Step 3: Compute the overall mean (\(\bar{X}_{\text{overall}}\)) by combining all data points from the three cities.
4. Step 4: Calculate the Sum of Squares Between Groups (SSB) using the formula: \[ SSB = \sum_{i=1}^3 n_i (\bar{X}_i - \bar{X}_{\text{overall}})^2 \] This measures the variation between the group means and the overall mean.
5. Step 5: Calculate the Sum of Squares Within Groups (SSW) by summing the squared deviations of each observation from its group mean: \[ SSW = \sum_{i=1}^3 \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2 \] Then, compute the Mean Squares Between (MSB) and Mean Squares Within (MSW) as: \[ MSB = \frac{SSB}{k - 1} \quad \text{and} \quad MSW = \frac{SSW}{N - k} \] where \(k=3\) is the number of groups and \(N\) is the total number of observations.
6. Step 6: Finally, calculate the test statistic \(F\) using the formula: \[ F = \frac{MSB}{MSW} \] This \(F\) value will be compared to the critical value from the \(F\) distribution with degrees of freedom \(df_1 = k - 1\) and \(df_2 = N - k\) to decide whether to reject the null hypothesis that all means are equal.

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more 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 here since we compare wages across three cities.

Test Statistic in ANOVA (F-statistic)

The F-statistic is the ratio of the variance between group means to the variance within the groups. A larger F-value suggests greater evidence against the null hypothesis. Calculating this involves computing group means, overall mean, and variances, then comparing the ratio to a critical value based on degrees of freedom.

Assumptions of ANOVA

ANOVA requires that samples are independent, populations are normally distributed, and population variances are equal (homogeneity of variance). These assumptions ensure the validity of the F-test. In this question, normality and equal variances are assumed, which justifies using ANOVA for comparing the three city wage means.