A professor finds the mean score of a special project due every week for the 21 students in his class. The mean scores are recorded in the table. He wants to make a control chart for the process mean and already knows the mean of the ranges, R=27.3. Find the centerline X, the upper central limit (UCL), and the lower central limit (LCL).

Question

A professor finds the mean score of a special project due every week for the 21 students in his class. The mean scores are recorded in the table. He wants to make a control chart for the process mean and already knows the mean of the ranges, R=27.3. Find the centerline X, the upper central limit (UCL), and the lower central limit (LCL).

Week: 1, 2, 3, 4, 5
Mean score: 86.2, 89.1, 76.8, 78.7, 83.5
A) x= 82.8600, UCL= 97.1948, LCL= 68.5252
B) x= 19.7286, UCL= 24.4515, LCL= 15.0057
C) x= 82.8600, UCL= 87.5829, LCL=78.1371
D) x= 82.8600, UCL= 98.6121, LCL= 67.1079

Answer 1

The first step is to find the overall mean score, which is the centerline X:

X = (86.2 + 89.1 + 76.8 + 78.7 + 83.5) / 5 = 82.86

The formula for the UCL and LCL are:

UCL = X + A*R
LCL = X - A*R

where A is the control chart constant, which depends on the sample size and the type of control chart being used. For this problem, since we have 21 samples, we can use the following table to find the value of A:

https://www.spcforexcel.com/knowledge/control-chart-basics/control-chart-constants

Using the table, we find that A = 1.73 for this situation.

Plugging in the values, we get:

UCL = 82.86 + 1.73*27.3 = 97.1948
LCL = 82.86 - 1.73*27.3 = 68.5252

Therefore, the answer is A) x= 82.8600, UCL= 97.1948, LCL= 68.5252.

Answer 2

To find the centerline (X), upper central limit (UCL), and lower central limit (LCL) for the control chart, we can use the formulas:

X = Mean of the sample means
UCL = X + 2.66 * R / sqrt(n)
LCL = X - 2.66 * R / sqrt(n)

Where:
R = Range of the sample means
n = Number of samples

Given:
Mean scores: 86.2, 89.1, 76.8, 78.7, 83.5
R = 27.3 (the mean of the ranges)
n = 5 (number of samples)

Step 1: Calculate the X (centerline):
X = (86.2 + 89.1 + 76.8 + 78.7 + 83.5) / 5
X = 413.3 / 5
X = 82.66

Step 2: Calculate the UCL and LCL:
UCL = 82.66 + 2.66 * 27.3 / sqrt(5)
UCL = 82.66 + 2.66 * 27.3 / 2.236
UCL = 82.66 + 2.66 * 12.208 / 2.236
UCL = 82.66 + 32.56 / 2.236
UCL = 82.66 + 14.57
UCL = 97.23

LCL = 82.66 - 2.66 * 27.3 / sqrt(5)
LCL = 82.66 - 2.66 * 27.3 / 2.236
LCL = 82.66 - 2.66 * 12.208 / 2.236
LCL = 82.66 - 32.56 / 2.236
LCL = 82.66 - 14.57
LCL = 68.09

Therefore, the correct answer is:
A) x= 82.6600, UCL= 97.23, LCL= 68.09