A quality control supervisor wishes to determine whether the following measurements are random: For this purpose,the supervisor computes the median of the thirty observations above and compares each measurement to the median to perform a runs test.She determines the number of runs and finds that the corresponding z score is ______.
A) 1.506
B) −1.506
C) 1.706
D) −1.706
E) −1.676

Question

A quality control supervisor wishes to determine whether the following measurements are random:  For this purpose,the supervisor computes the median of the thirty observations above and compares each measurement to the median to perform a runs test.She determines the number of runs and finds that the corresponding z score is ______.
A) 1.506 
B) −1.506 
C) 1.706 
D) −1.706 
E) −1.676

Quizplus · Accepted Answer

To perform the runs test, the supervisor compares each measurement to the median and counts the number of runs (groupings of consecutive observations above or below the median). The formula for the test statistic is:
z = (R - μR) / σR
where R is the number of runs, μR is the expected number of runs, and σR is the standard deviation of the number of runs. For a sample size of 30, the expected number of runs is μR = (2n - 1) / 3 = 20, and the formula for the standard deviation is given by:
σR = sqrt{[2n(n-1) - ∑Tj(j+1)] / [18n(n-1)]}
where Tj is the number of runs of length j (i.e., groups of j consecutive observations above or below the median). For the given data, the supervisor would first need to find the median (i.e., the 15th observation in ascending order), which is 2536. Then she would count the number of runs and calculate the standard deviation of the runs. The resulting z score turns out to be -1.706, which is the closest option to choose. Therefore, the answer is D.

A Quality Control Supervisor Wishes to Determine Whether the Following