Q: Given a process with $$\mu = 4$$ and $$\sigma = 3$$, and a sample size of $$n = 50$$, what are the 99% control limits for the mean?

$$\4$$ \pm 2.58 \times \frac{3}{\sqrt{50}}\$$

Question 1

Which of the following best describes the Central Limit Theorem as it applies to the sampling distribution of the mean?

Accepted Answer

The sampling distribution of the mean will be normal if the population is normal or if the sample size is sufficiently large.

Question 2

A manufacturer monitors the average HALT score of GPS chips using samples of size $$n = 20$$, with population mean $$\mu = 7$$ and standard deviation $$\sigma = 4. $$What is the standard error of the mean?

Accepted Answer

$$\$$\frac{4}{\sqrt{20}} \approx 0.89\$$

Question 3

Which of the following is a Type I error in the context of process control?

Accepted Answer

Taking action when no action is needed.

Question 4

If the control limits for a process are set too wide, what is the most likely consequence?

Accepted Answer

The chance of a Type II error increases.

Question 5

Given a process with $$\mu = 4$$ and $$\sigma = 3$$, and a sample size of $$n = 50$$, what are the 99% control limits for the mean?

Accepted Answer

$$\4$$ \pm 2.58 	imes \frac{3}{\sqrt{50}}\$$

Question 6

Why is it important to set control limits before looking at the data?

Accepted Answer

To ensure the limits are based on process parameters and not influenced by random sample variation.

Question 7

A call center monitors the mean call length with $$\mu = 4$$ minutes, $$\sigma = 3$$ minutes, and $$n = 50. If $$the observed mean is 6 minutes, is this likely to be a signal of a process change if the 99% control limits are used?

Accepted Answer

Yes, because 6 is outside the 99% control limits.

Question 8

Which chart would you use to monitor the variability (spread) of a process?

Accepted Answer

S-chart

Question 9

What is the main benefit of averaging when monitoring a process?

Accepted Answer

It makes the distribution of sample means more bell-shaped and reduces variance.

Question 10

If a process is monitored daily with a 5% chance of a Type I error each day, what is the approximate probability of at least one Type I error over 10 days?

Accepted Answer

40%

Question 11

Which of the following is NOT a best practice when using control charts?

Accepted Answer

Set control limits after looking at the data.

Question 12

What is the formula for the standard error of the mean?

Accepted Answer

$$\$$\frac{\sigma}{\sqrt{n}}\$$

Question 13

Which of the following statements about Type I and Type II errors is correct?

Accepted Answer

Type I error is taking action when no action is needed, and Type II error is failing to take action when needed.

Question 14

A process has $$\mu = 7$$, $$\sigma = 4$$, and $$n = 20. $$What is the probability that the sample mean is less than 6 or greater than 8 if the process is in control? (Assume normality and use $$\sigma_{\bar{X}} = 0.89.)$$

Accepted Answer

0.27

Question 15

Which of the following is a pitfall to avoid when using control charts?

Accepted Answer

Confusing the standard error of the mean with the standard deviation of the sample.

Question 16

Why does repeated testing increase the chance of a Type I error?

Accepted Answer

Because the probability of at least one false alarm increases with more tests.

Question 17

What is the main purpose of a control chart in quality monitoring?

Accepted Answer

To track the mean and/or variability of a process over time and detect signals of process changes.

Question 18

Which formula gives the 100(1 – $$\alpha$$)% control limits for the mean of a process?

Accepted Answer

$$\$$\mu \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\$$

Sampling Variation and Quality Control: Study Notes for Statistics for Business

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