CASE PROBLEM | Homework Helpers
CASE PROBLEM
EC 580, Instructor: Asiye Aydilek
Quality Associates, a consulting firm, advises its clients about sampling and
statistical procedures that can be used to control their manufacturing processes. In one
particular application, a client ave Quality Associates a sample of 800 observations taken
during a time in which that client’s process was operating satisfactorily. The sample
standard deviation for these data was 0.21 and hence, with so much data, the population
standard deviation was assumed to be 0.21. Quality Associates then suggested that
random samples of size 30 be taken periodically to monitor the process on an ongoing
basis. By analyzing the new samples, the client could quickly learn whether the process
was operating satisfactorily. When the process was not operating satisfactorily, corrective
action could be taken to eliminate the problem.
The design specification indicated the mean for the process should be 12. The
hypothesis test suggested by Quality Associates follows
H0
: µ=12
H1
: µ 12
Corrective action will be taken any time H0
is rejected.
The data set Quality’ contains data from four samples, each of size 30, collected at hourly
intervals during the first day of operation of the new statistical control procedure.
1- Conduct a hypothesis test for each sample at the 0.01 level of significance and determine
what action, if any, should be taken. Provide the test statistic and p-value for each test.
(Hint: Follow the steps below)
Step 1: Enter the data range (ex. A2:A31) into the =count cell formula in cell F4
Write Sample Size to the cell E4
Step 2: Enter the data range (ex. A2:A31) into the =average cell formula in cell F5
Write Sample Mean to the cell E5
Step 3: Enter the population standard deviation 0.21 into cell F6
Write population std.deviation to the cell E6
Step 4: Enter the hypothesized value for the population mean 12 into cell F8.
Write Hypothesized value to the cell E8.
Step 5: Enter the formula of the standard error =F6/sqrt(F4) into cell F10
Write Standard Error to the cell E10
Step 6: Enter the formula of the test statistic =(F5-F8)/F10 into cell F11
Write Test Statistic to the cell E11
Step 7: Enter the formula of p-value of lower tail =normsdist(F11) into cell F13
Write p-value (Lower Tail) to the cell E13
Step 8: Enter the formula of p-value of upper tail = 1- F13 into cell F14
Write p-value (Upper Tail) to the cell E14
Step 9: Enter the formula of p-value of two tail = 2*(min(F13,F14)) into cell F15
Write p-value (Two Tail) to the cell E15
Repeat the steps above for each sample. The formula for the second sample should be
written to G cells(replace F’s with G) and for the third sample to H cells and the fourth
sample to I cells. Also make the adjustments for the data range. For instance if the
second sample data range is B2:B31, the formulas in G cells should be using B2:B31
range.
Since our test is two-tailed test, use p-value (two tail) in cells (F15,G15,H15,I15) to make
the rejection decision for each sample.
If p-value>0.01 then the null hypothesis cannot be rejected.
If p-value<0.01 then the null hypothesis is rejected. In such a case, corrective action is
necessary.
2- Compute the standard deviation for each of the four samples. Does the assumption of
0.21 for the population standard deviation appear reasonable?
(Hint: Look at the range of the sample standard deviations for all four samples. If 0.21 is
in that range then the assumption of 0.21 is good)
3- Discuss the implications of changing the level of significance to larger value. What error
(Type 1 or Type 2) could increase if the level of significance is increased?
(Hint: Decide whether you will reject the null hypothesis more or less with a larger α
value.
Decide whether Rejecting more often means quicker or slower corrective action when
the process is out of control.
Decide we have higher of which error when there will be a higher error probability of
stopping the process and attempting corrective action when the process is operating
satisfactorily.)
Sample 1 Sample 2 Sample 3 Sample 4
11.55 11.62 11.91 12.02
11.62 11.69 11.36 12.02
11.52 11.59 11.75 12.05
11.75 11.82 11.95 12.18
11.90 11.97 12.14 12.11
11.64 11.71 11.72 12.07
11.80 11.87 11.61 12.05
12.03 12.10 11.85 11.64
11.94 12.01 12.16 12.39
11.92 11.99 11.91 11.65
12.13 12.20 12.12 12.11
12.09 12.16 11.61 11.90
11.93 12.00 12.21 12.22
12.21 12.28 11.56 11.88
12.32 12.39 11.95 12.03
11.93 12.00 12.01 12.35
11.85 11.92 12.06 12.09
11.76 11.83 11.76 11.77
12.16 12.23 11.82 12.20
11.77 11.84 12.12 11.79
12.00 12.07 11.60 12.30
12.04 12.11 11.95 12.27
11.98 12.05 11.96 12.29
12.30 12.37 12.22 12.47
12.18 12.25 11.75 12.03
11.97 12.04 11.96 12.17
12.17 12.24 11.95 11.94
11.85 11.92 11.89 11.97
12.30 12.37 11.88 12.23
12.15 12.22 11.93 12.25
