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, 57.1Little 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: Define the hypotheses for the ANOVA test. The null hypothesis $$H_0$$ states that the mean annual wages are the same for all three cities, i.e., $$\mu_{Ithaca} = \mu_{Little\ Rock} = \mu_{Madison}. $$The alternative hypothesis $$H_a$$ states that at least one city has a different mean wage. Step 2: Calculate the sample means and sample variances for each city using the given wage data. This involves summing the wages for each city and dividing by the number of observations to get 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 cities. This is done by summing all wages from all cities 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 cities), $$n_i is $$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:  
$$F = \frac{MSB}{MSW}$$  
Finally, compare the calculated F-statistic to the critical value from the F-distribution table at $$\alpha = 0.10$$ with degrees of freedom $$k-1$$ and $$N-k. If$$ $$F is $$greater than the critical value, reject the null hypothesis; otherwise, fail to reject it.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Analysis of Variance (ANOVA)

Null and Alternative Hypotheses

Assumption of Equal Population Variances