A manufacturer of ballpoint pens wishes to know if a machine designed to fill each individual pen with ink is operating properly.The median amount of ink should be 0.05 oz.Suppose 58 pens are randomly sampled from this particular machine, and none of the pens contains exactly 0.05 oz.Assuming the amount of ink in each pen is symmetrically distributed, conduct a Wilcoxon signed-rank test to see if the median amount of ink differs from 0.05 oz.The sum of positive differences (t+) from the sample is 502.Use a normal approximation and select the correct p-value and decision.(Use α = 0.05.)
A) p = 0.0183, reject H₀
B) p = 0.9817, fail to reject H₀
C) p = 0.0049, reject H₀
D) p = 0.1195, fail to reject H₀

Question

Quizplus · Accepted Answer

Since the sample size is less than 30 and the distribution is assumed to be symmetric, we can use the Wilcoxon signed-rank test. The null and alternative hypotheses are:

H₀: The median amount of ink is 0.05 oz.
H₁: The median amount of ink is not 0.05 oz.

We need to calculate the test statistic W, which is the smaller of the sum of positive signed ranks and the sum of negative signed ranks. In this case, all the differences are positive, so W is equal to the sum of the ranks of the positive differences.

Since we have a sample size of 58, we can use a normal approximation to calculate the p-value. The test statistic is:

W = 502

The expected value of W under the null hypothesis is:

E(W) = n(n+1)/4 = 58*59/4 = 851

The standard deviation of W is:

SD(W) = sqrt(n*(n+1)*(2n+1)/24) = sqrt(58*59*117/24) = 67.8

The z-score is:

z = (502 - 851) / 67.8 = -4.86

Using a standard normal distribution table or calculator, the p-value is found to be approximately 0.0049. Since this p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the median amount of ink differs from 0.05 oz. Therefore, the best choice is C.

A Manufacturer of Ballpoint Pens Wishes to Know If a Machine