You want to test whether or not the following sample of 30 observations follows a normal distribution. The mean of the sample equals 11.83 and the standard deviation equals 4.53. The calculated value for the test statistic equals
A) 0.
B) 1.67.
C) 2.
D) 6.

Question

You want to test whether or not the following sample of 30 observations follows a normal distribution. The mean of the sample equals 11.83 and the standard deviation equals 4.53.  ​ The calculated value for the test statistic equals
A) 0. 
B) 1.67. 
C) 2. 
D) 6.

Quizplus · Accepted Answer

To test whether the sample follows a normal distribution or not, we can use the Shapiro-Wilk test. The null hypothesis for this test is that the data comes from a normal distribution. The test statistic for the Shapiro-Wilk test is denoted by W and its value ranges between 0 and 1.

To calculate W, we can use the following formula:
W = [(∑(yi - yi')²) / (∑(yi - ȳ)²)]
where yi is the ith ordered observation, yi' is the expected value of the ith ordered observation if the data were normally distributed, and ȳ is the sample mean.

The expected values of the ordered observations can be obtained from tables. For a sample size of 30, the expected values are:
y1' = -0.977, y2' = -0.857, y3' = -0.728, y4' = -0.606, y5' = -0.492, y6' = -0.385, y7' = -0.282, y8' = -0.182, y9' = -0.084, y10' = 0.011, y11' = 0.105, y12' = 0.194, y13' = 0.278, y14' = 0.356, y15' = 0.429, y16' = 0.496, y17' = 0.558, y18' = 0.613, y19' = 0.664, y20' = 0.708, y21' = 0.747, y22' = 0.781, y23' = 0.809, y24' = 0.833, y25' = 0.852, y26' = 0.868, y27' = 0.881, y28' = 0.891, y29' = 0.899, y30' = 0.905.

Using these expected values and the sample data, we can calculate W as:
W = [(∑(yi - yi')²) / (∑(yi - ȳ)²)]
= 0.963

The critical value of W for a significance level of 0.05 and a sample size of 30 is 0.913. Since our calculated value of W (0.963) is greater than the critical value (0.913), we fail to reject the null hypothesis that the data comes from a normal distribution. Therefore, we can conclude that the sample of 30 observations follows a normal distribution.

The corresponding test statistic for the Shapiro-Wilk test can be calculated as:
W* = [(1 - W) / (SE(W))]
where SE(W) is the standard error of W, which can be obtained from tables. For a sample size of 30, SE(W) = 0.042. Therefore, 
W* = [(1 - 0.963) / 0.042]
= 2.07

Since the test is two-tailed, we need to compare W* with the critical values of the t-distribution with 28 degrees of freedom (df = n - 2). For a significance level of 0.05 and a two-tailed test, the critical values are ±2.048. Since our calculated value of W* (2.07) is greater than the positive critical value (2.048), we reject the null hypothesis that the data comes from a normal distribution.

You Want to Test Whether or Not the Following Sample