Question

background if you want to read:

Quality Improvement Partners (QIP) is a consulting firm that advises clients about sampling and statistical procedures that can be used to control their manufacturing processes. Your company has given QIP a sample of 800 observations taken during a time in which your process was operating satisfactorily.

The design specification indicates the mean measurement for the process should be 12. The sample standard deviation for these data was 0.21. With that much data, you can assume the population standard deviation is 0.21.

QIP has suggested that periodic random samples of size 30 be taken so you can monitor the process on an ongoing basis. By analyzing the new samples, you can quickly learn whether the process is operating satisfactorily or if it is operating differently than the expected mean. When the process is not operating satisfactorily, corrective action can be taken to attempt to eliminate the root cause(s) (assuming you know what they are). You will take corrective action whenever you reject the null hypothesis at the 0.01 significance level. Use this information to answer the questions in this Data Lab.

Sample 1	Sample 2	Sample 3	Sample 4
11.55	11.62	11.91	12.02
11.62	11.96	11.36	12.02
11.52	11.59	11.75	12.05
11.75	11.82	11.95	12.18
11.9	11.97	12.14	12.11
11.64	11.71	11.72	12.07
11.8	11.87	11.61	12.05
12.03	12.1	11.85	11.64
11.94	12.01	12.16	12.39
11.92	11.99	11.91	11.65
12.13	12.2	12.12	12.11
12.09	12.16	11.61	11.9
11.93	12	12.21	12.22
12.21	12.28	11.56	11.88
12.32	12.39	11.95	12.03
11.93	12	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	12.2
11.77	11.84	12.12	11.79
12	12.07	11.6	12.3
12.04	12.11	11.95	12.27
11.98	12.05	11.96	12.29
12.3	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
12.3	12.37	11.88	12.23
12.15	12.22	11.93	12.25

Question 1: If you conduct a hypothesis test on one of the four samples provided to determine if the process is operating satisfactorily, what kind should it be (given the information provided in the Background information provided)

a. Two sample z-test for the population mean

b. One sample t-test for the population mean

c. One sample z-test for the population mean

d. Two sample z-test for the population proportion

e. Two sample t-test for the population mean

f. One sample z-test for the population proportion

g. None of the above

Question 2: Using only the information given in the Background infromation for this DataLab, which of the following is the correct set of null and alternative hypothesis?

a. Ho:μ=12 and Ha:μ≥12

b. Ho:μ=12 and Ha:μ≤12

c. Ho:μ=12 and Ha:μ≠12

d. Ho:μ≠12 and Ha:μ=12

e. Ho:μ≠12 and Ha:μ≤12

f. None of the above

Question 3: Run the appropriate hypothesis test using StatCrunch and record the test-statistics (z or t) for each of the samples.

Sample 1:  (Round to 4 decimal places)

Sample 2:  (Round to 4 decimal places)

Sample 3:  (Round to 4 decimal places)

Sample 4:  (Round to 4 decimal places)

Question 4: Run the appropriate hypothesis test using StatCrunch and record the p-values for each sample

Sample 1:  (Round to 4 decimal places)

Sample 2:  (Round to 4 decimal places)

Sample 3:  (Round to 4 decimal places)

Sample 4:  (Round to 4 decimal places)

Question 5: Compute the standard deviation of each sample:

Sample 1:  (Round to 4 decimal places)

Sample 2:  (Round to 4 decimal places)

Sample 3:  (Round to 4 decimal places)

Sample 4:  (Round to 4 decimal places)

Question 6: Based on your sample standard deviation results (previous question), is the assumption of .21 for the population standard deviation appear reasonable?
a. Yes

b. No

c. It's impossible to say

Question 7: Based on your hypothesis test results, which of the following sample(s) would lead you to take corrective action. CHECK ALL THAT APPLY. (You only get full credit if you have all the appropriate boxes checked AND none of the inappropriate boxes checked.)

Sample 1

Sample 2

Sample 3

Sample 4

Compute limits for the sample mean ¯x around μ=12 such that, as long as a new sample mean is within those limits, the process will be considered to be operating satisfactorily. If ¯x exceeds the upper limit or if ¯x is below the lower limit, corrective action will be taken. For quality control purposes, these limits are referred to as lower and upper control limits, respectively:
Upper Control Limit =  (Round answers to 4 decimal places)

Lower Control Limit =  (Round answers to 4 decimal places)

Question 9: What would happen if you changed the level of significance to a larger value?
a. Corrective action will be taken more often

b. Corrective action will be taken less often

c. It will have no effect on how often we would take corrective action

Question 10: For a sample(s) in this data set where corrective action is warranted based on the test output, which of the following statements is most accurate?
a. It is impossible to know if the mean will be inside or outside of the confidence interval just found.

b. The mean of the sample is inside the confidence interval just found

c. The mean of the sample is outside the confidence interval just found

d. There is a 99% chance that the mean of the sample is somewhere between the lower control limit and the upper control limit.

I get question 1,2,5, and 6 but not the rest.

PLEASE HELP ME!

Answer 1

Question 1,2,3 & 4:

If you conduct a hypothesis test on one of the four samples provided to determine if the process is operating satisfactorily, what kind should it be (given the information provided in the Background information provided). Using only the information given in the Background information for this DataLab, which of the following is the correct set of null and alternative hypothesis? Run the appropriate hypothesis test using StatCrunch and record the test-statistics (z or t) for each of the samples and record the p-values for each sample.

We will be using one sample T-test refers to a univariate hypothesis test based on t-statistic, wherein the mean is known, and population variance is approximated from the sample and the sample size is not more than 30. To test the null hypothesis that the mean the of sample is equal to 12 and the alternate hypothesis that the mean is not equal to 12.

Level of significance = alpha = 0.01

Test statistic of one sample is given as

where,

\bar{X} is the mean of the sample, \mu is the hypothesized mean, s is the sample standard deviation and n is the sample size.

Using EXCEL I have calculated, sample statistic and p value

	Sample 1	Sample 2	Sample 3	Sample 4
Mean	11.95867	12.03767	11.889	12.08133
std	0.220356	0.211378	0.207171	0.206109
n	30	30	30	30
S.E	0.040231	0.038592	0.037824	0.03763
t stats	-1.02739	0.976019	-2.93464	2.161386
p	0.31273	0.337127	0.006469	0.039059

The formula used for the caluclation are as follows

	Sample 1	Sample 2	Sample 3	Sample 4
Mean	=AVERAGE(B2:B31)	=AVERAGE(C2:C31)	=AVERAGE(D2:D31)	=AVERAGE(E2:E31)
std	=STDEV.S(B2:B31)	=STDEV.S(C2:C31)	=STDEV.S(D2:D31)	=STDEV.S(E2:E31)
n	30	30	30	30
S.E	=B34/SQRT(B35)	=C34/SQRT(C35)	=D34/SQRT(D35)	=E34/SQRT(E35)
t	=(B33-12)/B36	=(C33-12)/C36	=(D33-12)/D36	=(E33-12)/E36
p	=T.DIST.2T(ABS(B37),29)	=T.DIST.2T(ABS(C37),29)	=T.DIST.2T(ABS(D37),29)	=T.DIST.2T(ABS(E37),29)

Question 7: Based on your hypothesis test results, which of the following sample(s) would lead you to take corrective action. CHECK ALL THAT APPLY. (You only get full credit if you have all the appropriate boxes checked AND none of the inappropriate boxes checked.)

At the significance level of 0.01, only Sample 3 rejects the null hypothesis as p-value(0.006) is less than 0.01. Thus accepting the alternate hypothesis for sample 3 that its mean is different from 12.

For sample 1, sample 2 and sample 4, the p-value is more than 0.01, thus Null hypothesis is accepted and we conclude that the sample means of 1,2, and 4 samples are equal to 12.

Compute limits for the sample mean ¯x around μ=12 such that, as long as a new sample mean is within those limits, the process will be considered to be operating satisfactorily. If ¯x exceeds the upper limit or if ¯x is below the lower limit, corrective action will be taken. For quality control purposes, these limits are referred to as lower and upper control limits, respectively:

Confidence limit at alpha = 0.01, can be calculated using the following formula

with df = n-1 = 30-1 = 29

First we need to find the t value alpha/2 = 0.01/2 = 0.005 and df = 29, from the table we get t =2.76

Upper Control Limit = 12 + 0.11 = 12.1100
Lower Control Limit = 12-0.11 = 11.8900

Question 9: What would happen if you changed the level of significance to a larger value?

Corrective action will be taken more often. For eg, we increase alpha =0.05, then sample 4 will also need corrective action along with sample 3.

Question 10: For a sample(s) in this data set where corrective action is warranted based on the test output, which of the following statements is most accurate?

b. The mean of the sample is inside the confidence interval just found

d. There is a 99% chance that the mean of the sample is somewhere between the lower control limit and the upper control limit.

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