Question

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, 57.1Little 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, 31.5Madison, 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, 66.2Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.

Accepted Answer

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. 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. Step 3: Compute the overall mean ($$\bar{X}_{	ext{overall}}) by $$combining all data points from the three cities. Step 4: Calculate the Sum of Squares Between Groups (SSB) using the formula:  
$$
SSB = \sum_{i=1}^3 n_i (\bar{X}_i - \bar{X}_{	ext{overall}})^2
$$  
This measures the variation between the group means and the overall mean. 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:  
$$
MSB = \frac{SSB}{k - 1} \quad 	ext{and} \quad MSW = \frac{SSW}{N - k}
$$  
where $$k=3 is $$the number of groups and $$N is $$the total number of observations. 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.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Analysis of Variance (ANOVA)

Test Statistic in ANOVA (F-statistic)

Assumptions of ANOVA