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STAT 2593 Final Examination: Quality Control in Manufacturing, Exams of Statistics

Senate of Serampore College (University)Statistics

The final examination questions for a statistics course focused on quality control in manufacturing. The questions involve calculating control limits for x-chart and s-chart based on data collected during a run-in period, determining if samples indicate statistical control, and analyzing data from experiments comparing two methods for determining residual chlorine content in sewage effluents and the relationship between smoking habits and household income.

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Department of Mathematics & Statistics

STAT 2593 Final Examination

December 10, 1997

TIME: 3 hours. Tota l m ar ks: 74.

SHOW ALL WORK!

1. (a) During a 40 hour “run-in” period, samples consisting of 4 power supply units 16 marks

were selected once each hour from an assembly line, and the high-voltage output

from each unit was determined. The means and standard deviations of each of

the 40 samples were determined and entered into columns of a Minitab work-

sheet. Summary statistics for these columns were then calculated via the Minitab

“DESCRIBE” command with the following results:

MTB > desc c5 c6

Variable N Mean Median TrMean StDev SEMean

xbar 40 2999.4 2999.4 2999.5 8.5 1.4

s 40 19.034 18.347 19.122 5.616 0.888

Variable Min Max Q1 Q3

xbar 2979.1 3018.1 2993.3 3004.1

s 6.754 30.424 15.859 24.597

Determine 3 SD control limits for (i) an ¯

X-chart and (ii) an S-chart for the

process, assuming that the process was in statistical control during the the 40

hour run-in period.

(b) Suppose that for a similar assembly line in a different company the control

limits are UCL = 3296 volts and LCL = 3224 volts for the ¯

X-chart, and UCL

= 50.15 volts and LCL = 0 volts for the S-chart. Three of the hourly samples

that were selected from the assembly line were entered into columns of a Minitab

worksheet and DESCribed, with the following results:

MTB > desc c1-c3

Variable N Mean Median TrMean StDev SEMean

sample.1 4 3197.1 3197.4 3197.1 21.9 11.0

sample.2 4 3263.0 3256.8 3263.0 23.5 11.8

sample.3 4 3277.2 3279.7 3277.2 80.3 40.1

Variable Min Max Q1 Q3

sample.1 3173.3 3220.2 3176.0 3217.8

sample.2 3242.6 3295.5 3244.3 3287.7

sample.3 3182.7 3367.0 3198.4 3353.7

......(continuedover page)

Partial preview of the text

Download STAT 2593 Final Examination: Quality Control in Manufacturing and more Exams Statistics in PDF only on Docsity!

Department of Mathematics & Statistics

STAT 2593 Final Examination

December 10, 1997

TIME: 3 hours. Total marks: 74.

SHOW ALL WORK!

1. (a) During a 40 hour “run-in” period, samples consisting of 4 power supply units 16 marks

were selected once each hour from an assembly line, and the high-voltage output

from each unit was determined. The means and standard deviations of each of

the 40 samples were determined and entered into columns of a Minitab work-

sheet. Summary statistics for these columns were then calculated via the Minitab

“DESCRIBE” command with the following results:

MTB > desc c5 c Variable N Mean Median TrMean StDev SEMean xbar 40 2999.4 2999.4 2999.5 8.5 1. s 40 19.034 18.347 19.122 5.616 0.

Variable Min Max Q1 Q xbar 2979.1 3018.1 2993.3 3004. s 6.754 30.424 15.859 24.

Determine 3 SD control limits for (i) an X¯-chart and (ii) an S-chart for the

process, assuming that the process was in statistical control during the the 40

hour run-in period.

(b) Suppose that for a similar assembly line in a different company the control

limits are UCL = 3296 volts and LCL = 3224 volts for the X¯-chart, and UCL

= 50.15 volts and LCL = 0 volts for the S-chart. Three of the hourly samples

that were selected from the assembly line were entered into columns of a Minitab

worksheet and DESCribed, with the following results:

MTB > desc c1-c Variable N Mean Median TrMean StDev SEMean sample.1 4 3197.1 3197.4 3197.1 21.9 11. sample.2 4 3263.0 3256.8 3263.0 23.5 11. sample.3 4 3277.2 3279.7 3277.2 80.3 40.

Variable Min Max Q1 Q sample.1 3173.3 3220.2 3176.0 3217. sample.2 3242.6 3295.5 3244.3 3287. sample.3 3182.7 3367.0 3198.4 3353.

...... (continued over page)

For each of the three samples, determine if there is an indication that the system

was out of statistical control with respect to either location or spread at the time

the sample was collected, giving a brief reason for your answer.

(c) Suppose that for the assembly line referred to in part (b) the process mean shifts

from its “in control value” of 3260 to 3210. What is the probability that this

change will be detected via the X¯-chart the next time that a sample is taken?

(d) Suppose that the mean of the process changes in such a way that the probability

that this change will be detected, via the X¯-chart, is 0.27 each time a sample is

taken.

(i) What is the probability that exactly 5 samples are taken before the change

is detected? (I. e. the change is detected at the 5th^ sample after the change

occurred.)

(ii) How may samples do you expect to be taken until the change is detected?

2. In an experiment to compare two different methods (“MSI” and “SIB”) for determin- 10 marks

ing residual chlorine content in sewage effluents, 8 specimens of water were measured

using each of the methods. The measurements were recorded in a Minitab worksheet.

Following there are two analyses of these data (parts of which have been obliterated).

One of these analyses is correct; the other is incorrect.

MTB > #----------------------------------------------------------

MTB > # Analysis number 1: MTB > twos c1 c2; SUBC> pool; SUBC> alte **.

Twosample T for MSI vs SIB N Mean StDev SE Mean MSI 8 5.02 4.22 1. SIB 8 5.43 4.12 1.

95% C.I. for mu MSI - mu SIB: ( ****, ****) T-Test mu MSI = mu SIB (vs *****): T= **** P=**** DF= ** Both use Pooled StDev = 4. MTB > #----------------------------------------------------------

...... (continued over page)

ChiSq = 0.669 + 6.326 + 0.348 + 1.228 + 0.354 + 3.059 + 0.849 + ***** + 1.649 + (v) + 3.889 + 4.525 + 0.131 + 0.516 + 0.484 = 25. df = (vi), p = *****

(a) Supply the missing entries in the Minitab output, labelled (i), (ii),... , (vi).

(Note that other entries, which you are NOT requested to supply, have been

obliterated as well.)

(b) State the appropriate null and alternative hypotheses clearly.

(c) On the basis of the Minitab output (including the bits supplied by you) bracket

the p-value for the hypothesis being tested,

(d) Summarize briefly in words, with reference to the “standard” significance levels

(i.e. 0.10, 0.05, and 0.01) what you conclude from this analysis.

4. An experiment was performed to compare the fracture toughness of high-purity 18 Ni 10 marks

maraging steel with commercial purity steel of the same type. Thirty-two specimens

of steel of each type were measured, and the data were entered into columns of a

Minitab worksheet. Following there are two analyses of these data (parts of which

have been obliterated). One of these analyses is correct; the other is incorrect.

MTB > #----------------------------------------------------------

MTB > # Analysis number 1: MTB > twos c1 c2; SUBC> pool.

Twosample T for high vs commerc. N Mean StDev SE Mean high 32 65.01 1.60 0. commerc. 32 60.64 1.38 0.

95% C.I. for mu high - mu commerc.: ( ****, ****) T-Test mu high = mu commerc. (vs not =): T= **** P=**** DF= ** Both use Pooled StDev = 1. MTB > #----------------------------------------------------------

MTB > #---------------------------------------------------------- MTB > # Analysis number 2: MTB > let c3 = c1 - c MTB > tint c MTB > name c3 ’hi-com’

Variable N Mean StDev SE Mean 95.0 % C.I. hi-com 32 4.369 2.103 0.372 ( ****, ****) MTB > #----------------------------------------------------------

...... (continued over page)

(a) Which is the correct analysis? Give a brief reason.

(b) Using the correct analysis, (you will need to supply some of the obliterated

items) find a 95% confidence interval for the population mean difference in frac-

ture toughness between the two grades of steel.

(c) Suppose it is claimed that the high-purity steel is 10 units tougher on average

than the commercial-purity steel. On the basis of your confidence interval do

you think this claim is true?

5. In a clinical trial on the effectiveness of two drugs (ticrynafen and hydrochlorithiazide) 9 marks

in the treatment of high blood pressure each of the two drugs was given at either a

high or a low dosage level to separate groups of subjects for 6 weeks. The response

was recorded as drop (baseline minus final value) in systolic blood pressure (mm Hg)

for each subject. The data were analyzed in Minitab, producing the following output,

parts of which have been obliterated:

MTB > retr ’f:drugs’ MTB > aovo c1-c

Analysis of Variance Source DF SS MS F p Factor (i) (iv) 222.9 (vi) ***** Error (ii) 2119.6 (v) Total 27 2788. Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----+---------+---------+---------+-- Tic-lo 8 13.762 8.221 (------------) Tic-hi 7 20.629 6.480 (-------------) Hyd-lo (iii) 5.620 7.909 (---------------) Hyd-hi 8 15.450 12.786 (-----------) ----+---------+---------+---------+-- Pooled StDev = 9.398 0 10 20 30

STAT 2593 Final Examination: Quality Control in Manufacturing, Exams of Statistics

Related documents

Partial preview of the text

Download STAT 2593 Final Examination: Quality Control in Manufacturing and more Exams Statistics in PDF only on Docsity!

Department of Mathematics & Statistics

STAT 2593 Final Examination

December 10, 1997

TIME: 3 hours. Total marks: 74.

SHOW ALL WORK!

1. (a) During a 40 hour “run-in” period, samples consisting of 4 power supply units 16 marks

were selected once each hour from an assembly line, and the high-voltage output

from each unit was determined. The means and standard deviations of each of

the 40 samples were determined and entered into columns of a Minitab work-

sheet. Summary statistics for these columns were then calculated via the Minitab

“DESCRIBE” command with the following results:

Determine 3 SD control limits for (i) an X¯-chart and (ii) an S-chart for the

process, assuming that the process was in statistical control during the the 40

hour run-in period.

(b) Suppose that for a similar assembly line in a different company the control

limits are UCL = 3296 volts and LCL = 3224 volts for the X¯-chart, and UCL

= 50.15 volts and LCL = 0 volts for the S-chart. Three of the hourly samples

that were selected from the assembly line were entered into columns of a Minitab

worksheet and DESCribed, with the following results:

...... (continued over page)

For each of the three samples, determine if there is an indication that the system

was out of statistical control with respect to either location or spread at the time

the sample was collected, giving a brief reason for your answer.

(c) Suppose that for the assembly line referred to in part (b) the process mean shifts

from its “in control value” of 3260 to 3210. What is the probability that this

change will be detected via the X¯-chart the next time that a sample is taken?

(d) Suppose that the mean of the process changes in such a way that the probability

that this change will be detected, via the X¯-chart, is 0.27 each time a sample is

taken.

(i) What is the probability that exactly 5 samples are taken before the change

is detected? (I. e. the change is detected at the 5th^ sample after the change

occurred.)

(ii) How may samples do you expect to be taken until the change is detected?

2. In an experiment to compare two different methods (“MSI” and “SIB”) for determin- 10 marks

ing residual chlorine content in sewage effluents, 8 specimens of water were measured

using each of the methods. The measurements were recorded in a Minitab worksheet.

Following there are two analyses of these data (parts of which have been obliterated).

One of these analyses is correct; the other is incorrect.

MTB > #----------------------------------------------------------

...... (continued over page)

(a) Supply the missing entries in the Minitab output, labelled (i), (ii),... , (vi).

(Note that other entries, which you are NOT requested to supply, have been

obliterated as well.)

(b) State the appropriate null and alternative hypotheses clearly.

(c) On the basis of the Minitab output (including the bits supplied by you) bracket

the p-value for the hypothesis being tested,

(d) Summarize briefly in words, with reference to the “standard” significance levels

(i.e. 0.10, 0.05, and 0.01) what you conclude from this analysis.

4. An experiment was performed to compare the fracture toughness of high-purity 18 Ni 10 marks

maraging steel with commercial purity steel of the same type. Thirty-two specimens

of steel of each type were measured, and the data were entered into columns of a

Minitab worksheet. Following there are two analyses of these data (parts of which

have been obliterated). One of these analyses is correct; the other is incorrect.

MTB > #----------------------------------------------------------

...... (continued over page)

(a) Which is the correct analysis? Give a brief reason.

(b) Using the correct analysis, (you will need to supply some of the obliterated

items) find a 95% confidence interval for the population mean difference in frac-

ture toughness between the two grades of steel.

(c) Suppose it is claimed that the high-purity steel is 10 units tougher on average

than the commercial-purity steel. On the basis of your confidence interval do

you think this claim is true?

5. In a clinical trial on the effectiveness of two drugs (ticrynafen and hydrochlorithiazide) 9 marks

in the treatment of high blood pressure each of the two drugs was given at either a

high or a low dosage level to separate groups of subjects for 6 weeks. The response

was recorded as drop (baseline minus final value) in systolic blood pressure (mm Hg)

for each subject. The data were analyzed in Minitab, producing the following output,

parts of which have been obliterated:

(a) State (clearly) the hypotheses being tested.

(b) Supply the missing numbers from the locations labelled (i), (ii),... (vi) in the

above Minitab output.

(c) State your decision about the test at the 0.05 and 0.01 significance levels.

(d) Explain briefly in words what your decisions mean.

6. (Continued.)

(a) Is the intercept (constant term) of the true regression line equal to 0? Explain

briefly.