Question

A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that the process standard deviation is two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below.

Weight
Day Package 1 Package 2 Package 3 Package 4
Monday 23 22 23 24
Tuesday 23 21 19 21
Wednesday 20 19 20 21
Thursday 18 19 20 19
Friday 18 20 22 20

A. Calculate upper 2-sigma x-bar chart control limit that allow for natural variations

Answer 1

The upper 2-sigma control limits for the x-bar chart are:

Monday: 26.5 ounces

Tuesday: 25.0 ounces

Wednesday: 24.0 ounces

Thursday: 23.0 ounces

Friday: 24.0 ounces

Calculating Upper Control Limit for x-bar Chart:

Given:

Process standard deviation (σ) = 2 ounces

Sample size (n) = 4 packages per day

Data for Monday to Friday

Steps:

Calculate the mean (x) for each day:

Monday: (23 + 22 + 23 + 24) / 4 = 22.5 ounces

Tuesday: (23 + 21 + 19 + 21) / 4 = 21.0 ounces

Wednesday: (20 + 19 + 20 + 21) / 4 = 20.0 ounces

Thursday: (18 + 19 + 20 + 19) / 4 = 19.0 ounces

Friday: (18 + 20 + 22 + 20) / 4 = 20.0 ounces

Calculate the upper 2-sigma control limit:

Upper control limit (UCL) = x + 2 * σ / √n

For Monday: UCL = 22.5 + 2 * 2 / √4 = 26.5 ounces

For Tuesday: UCL = 21.0 + 2 * 2 / √4 = 25.0 ounces

For Wednesday: UCL = 20.0 + 2 * 2 / √4 = 24.0 ounces

For Thursday: UCL = 19.0 + 2 * 2 / √4 = 23.0 ounces

For Friday: UCL = 20.0 + 2 * 2 / √4 = 24.0 ounces

Therefore, the upper 2-sigma control limits for the x-bar chart are:

Monday: 26.5 ounces

Tuesday: 25.0 ounces

Wednesday: 24.0 ounces

Thursday: 23.0 ounces

Friday: 24.0 ounces

Answer 2

The upper 2-sigma x-bar chart control limit allows for natural variations and can be calculated using the overall sample mean, the process standard deviation, and the size of the samples. For the given data, the upper 2-sigma control limit is 22.6 ounces.

Step-by-step explanation:

To construct a sample mean chart, also known as an x-bar chart, for controlling a packaging process, we will need to calculate the upper 2-sigma control limit. The question provides us with a known process standard deviation (sigma) of two ounces. To calculate the upper control limit (UCL), we use the following formula:

UCL = x-double bar + Z * (sigma/sqrt(n))

Where x-double bar is the overall sample mean, Z is the Z-score corresponding to the desired confidence level (for 2-sigma we typically use a Z-score of approximately 2 since it corresponds to approximately 95% of the data assuming a normal distribution), sigma is the standard deviation, and n is the sample size.

Since the process standard deviation is known to be two ounces, we directly use this value in our formula. However, to calculate the x-double bar we need the mean of all daily means.

The mean package weights for the days are as follows:

• Monday's mean = (23+22+23+24)/4 = 23 ounces
• Tuesday's mean = (23+21+19+21)/4 = 21 ounces
• Wednesday's mean = (20+19+20+21)/4 = 20 ounces
• Thursday's mean = (18+19+20+19)/4 = 19 ounces
• Friday's mean = (18+20+22+20)/4 = 20 ounces

The overall sample mean (x-double bar) is the average of these daily means: (23+21+20+19+20)/5 = 20.6 ounces.

Now, we can use Z = 2 for the 2-sigma:

UCL = 20.6 + 2 * (2/sqrt(4)) = 20.6 + 2 ounces = 22.6 ounces

This upper 2-sigma control limit allows for natural variations within the process.