Question

Randomly select the heights of at least 15 of the students in your class.

a. Develop a control chart and plot the heights on the chart

b. Which chart should you use?

c. Is this process in control?

I have organized the heights of the 15 students below from shortest to tallest. Please show all work and complete in Excel! If you could show the formulas also, that would be great. Thank you!

Student	Height (Inches)
1	59
2	61
3	62
4	64
5	65
6	65
7	66
8	69
9	70
10	70
11	71
12	72
13	72
14	74
15	77

Answer 1

a. Objective: To monitor the height of students in class, based on a sample of 15 and to determine whether the process is in control.

b. Since the variable of interest, Height is a numeric (continuous) variable, the appropriate measure to summarize the sample would be the sample mean $\overline{X}$ . Hence, the appropriate chart used to determine whether the process would be in control would be the $\overline{X}$ chart.

In order to construct an $\overline{X}$ chart, we first need to obtain the control limits, which can be computed using the formula:

XTAR

where, for the given sample,

Xi = 1,2,3,.., 15

X- ΣΧ.

_59+61 +...+" – 678 15

and Range $\overline{R}=$ Largest observation - Smallest observation

= 77 - 59

=18

The process can be considered to be in control if all the observations in the sample lie within the control limits.

Using excel,

Student Height (Inches) auw. WWU UN -7 X bar AE EAVERAGE(B2:B16 15Rx 1C AVERAGE(number1, (number2], ...)

Student Height (Inches) X bar 67.8 15Rx 1C =MAX(B2:B16)-MIN(B2:B16 MIN(number1, [number2], ...)

A₂ the control chart constant can be obtained from the table:

Table of Control Chart Constants R Chart Constants S Chart Constants X-bar Chart Constants for sigma estimate Sample Size = m A2 A3 d2 D3 D4 B3 B4 3 3.267 0 2.574 2.282 0 0 2.114 10 1.880 2.659 1.1280 1.0231.954 1.6930 0.729 1.628 2.0590 0.5771 1.427 0.483 1.287 2.534 0.419 1.182 2.704 0.076 0.373 1.0992.847 0.136 0.337 1.032 2.970 0.184 0.308 0.975 0.223 0.285 0.927 3.173 0.256 0.266 0.886 3.258 0.283 0.249 0.850 3.336 0.307 0.235 0.817 3.407 0.328 L0.223 10.789 3.472 L0.347 0.212 0.763 3.532 0.363 0 2031 To 739 3588 In 378 L-9999 13.078 2.004 1.924 1.864 | 1.816 | 1.777 1.744 | 1.717 1.693 1.672 1.653 1.637 1622 .267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716 1.679 | 1.646 1.618 1.594 1.572 | 1.552 0.030 0.118 0.185 0.239 1 0.284 0.321 0.354 0.382 10.406 0.428 0.448 166

Substituting the values,

The Lower Control Limit is obtained as:

Α в Student Height (Inches) 1 D LCL UCL =B17-B19*B18 1 59 2 61 62 i in mm O mto mt 72 13 14 6 15 7 X bar 8 R 9 A2 67.8 0.223

The Upper Control Limit is obtained as:

E Α Student Height (Inches) C D LCL UCL 63.786 =B17+B19*B18 67.8 - X bar BR Ə A2 0.223

The graph is constructed as:

Home Insert Page Layout Formulas Data Review View 7 38 TP 2 il XXX PivotTable Table Picture Clip Shapes SmartArt Column Line Pie Bar Area Scatt Art Tables Illustrations 2-D Line B1 А Student Height (Inches) 3-D Line onmt innmm 9 i All Chart Types... fax Height (Inches) C D LCL UCL 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814 63.786 71.814) -7 X bar 00 67.8 18 0.223 9 A2

Student Height (Inches) 77 72 72 71 70 70 71.814 69 LCL 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 63.786 UCL 1 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 71.814 64 65 65 66 Height (Inches) Height (in inches) 63.786 LCL 59672 +UCL 14 115 X bar R A2 77 67.8 18 0.223 I. | 5011 1 2 3 4 5 6 9 10 11 12 13 14 15 7 8 Student

c. From the control chart obtained,

We find that the observations corresponding to the samples 1,2,3,14 and 15 lie outside the control limits. Hence, we fail to conclude that the process is in control based on the given sample.