Problem 10-14 Analyze the data in the problems listed below using median and up/

Question

Problem 10-14

Analyze the data in the problems listed below using median and up/down run tests with z = ± 2.

a. Given the following run test results of process output, what do the results of the run tests suggest about the proces

non-random

random

b.Twenty means were plotted on a control chart. An analyst counted 10 runs above/below the median, and 11 up/down runs. What do the results suggest about the process?

random

non-random

c. The following data are the number of defects per spool of cable.

d. The postmaster of a small western town receives a certain number of complaints each day about mail delivery.

Test z-score Median +2.54 Up/Down +2.13

Explanation / Answer

(a)

Both the z-scores are more than the +/- 2.0 limit. So, for both of them, non-random

(b)

Median method

N=20
Number of observed runs = 10
Expected number of runs = (N/2) + 1 = (20/2) + 1 = 11
Variance = (N/2)*((N/2) - 1)/(N - 1) = 10*(10 - 1)/(20 - 1) = 4.736; Std. Deviation = SQRT(Variance) = 2.176
z-score = (Observed runs - Expected runs) / Std. Dev = (10 - 11)/2.176 = -0.46 (random since it is within the range of +/- 2.0)

Up/ down method

N=20
Number of observed runs = 11
Expected number of runs = (2N - 1)/3 = (2*20 - 1) /3 = 13
Variance = (16N - 29)/90 = (16*20 - 29) / 90 = 3.23; Std. Deviation = SQRT(Variance) = 1.80
z-score = (Observed runs - Expected runs) / Std. Dev = (11 - 13)/1.80 = -1.11  (random since it is within the range of +/- 2.0)

(c)

Median method

For the data given, median = 1.0

N=14
Number of observed runs = 6
Expected number of runs = (N/2) + 1 = (14/2) + 1 = 8
Variance = (N/2)*((N/2) - 1)/(N - 1) = 7*(7 - 1)/(14 - 1) = 3.23; Std. Deviation = SQRT(Variance) = 1.80
z-score = (Observed runs - Expected runs) / Std. Dev = (6 - 8)/1.80 = -1.11 (random since it is within the range of +/- 2.0)

Up/ down method

N=14
Number of observed runs = 10
Expected number of runs = (2N - 1)/3 = (2*14 - 1) /3 = 9
Variance = (16N - 29)/90 = (16*14 - 29) / 90 = 2.17; Std. Deviation = SQRT(Variance) = 1.47
z-score = (Observed runs - Expected runs) / Std. Dev = (10 - 9)/1.80 = 0.556 (random since it is within the range of +/- 2.0)

(d)

Median Method

Median of the data = 7.5

N=14
Number of observed runs = 6
Expected number of runs = (N/2) + 1 = (14/2) + 1 = 8
Variance = (N/2)*((N/2) - 1)/(N - 1) = 7*(7 - 1)/(14 - 1) = 3.23; Std. Deviation = SQRT(Variance) = 1.80
z-score = (Observed runs - Expected runs) / Std. Dev = (6 - 8)/1.80 = -1.11 (random since it is within the range of +/- 2.0)

Up/ down method

N=14
Number of observed runs = 7
Expected number of runs = (2N - 1)/3 = (2*14 - 1) /3 = 9
Variance = (16N - 29)/90 = (16*14 - 29) / 90 = 2.17; Std. Deviation = SQRT(Variance) = 1.47
z-score = (Observed runs - Expected runs) / Std. Dev = (7 - 9)/1.80 = -1.11 (random since it is within the range of +/- 2.0)  

OBSERVATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Number of defects 1 3 1 0 1 3 2 0 2 1 3 1 2 0 Above/ below - A - B - A A B A - A - A B Runs 1 2 3 4 5 6

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