Question

An automobile manufacturing company examined the cars of a particular model to identify the number of defects during the final inspection stage. The total number of inspected carswere recorded for last 35 days along with the number of defects. The results are given in the following table:

Days Total Inspected Cars Number of Defects
1 45 6
2 35 2
3 40 1
4 30 2
5 40 5
6 25 1
7 35 3
8 30 4
9 30 6
10 40 2
11 35 11
12 35 1
13 25 3
14 40 7
15 45 2
16 40 6
17 30 5
18 35 2
19 30 7
20 45 3
21 28 1
22 38 4
23 33 11
24 33 1
25 43 4
26 38 2
27 38 5
28 28 4
29 43 6
30 48 1
31 43 3
32 33 4
33 38 3
34 33 2
35 48 5

Construct a suitable control chart for the number of defects to check whether the process is under statistical control or not. Also plot the revised control charts, if necessary.

11 Mar 2022
Answer :
Word Count : 656
Alright — since this is a *number of defects per inspection* problem, we’ll construct a u-chart because the sample sizes (cars inspected per day) are *not constant*. --- ## Step 1 — Formula for u-chart For a u-chart: $$ u_i = \frac{\text{Number of Defects in Sample } i}{\text{Sample Size in } i} $$ $$ \bar{u} = \frac{\text{Total Defects in All Samples}}{\text{Total Units Inspected}} $$ Control limits for each day: $$ UCL_i = \bar{u} + 3\sqrt{\frac{\bar{u}}{n_i}} $$ $$ LCL_i = \bar{u} - 3\sqrt{\frac{\bar{u}}{n_i}} $$ If LCL < 0, take LCL = 0. --- ## Step 2 — Compute totals ### Total inspected cars Let’s add them step-by-step (avoiding mistakes): 45 ___ _________ ____ ____ _______ _______ ______.
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Related Questions

A Company wants to maintain the quality of bottling process which uses a  particular brand of machine to fill 100 ml sanitizer spray bottle. During each  shift, a sample of 10 bottles is selected (2 hours apart) and the volume of the each filled sanitizer spray bottle (in ml) is determined. In this regards, total 25 subgroups consisting of a sample of 10 bottles in each subgroup were selected. The following table lists the measurements from 25 consecutive shifts:

Sample No. Obs. 1 Obs. 2 Obs. 3 Obs. 4 Obs. 5 Obs. 6 Obs. 7 Obs. 8 Obs. 9 Obs. 10
1 99.46 100.12 99.73 99.56 99.46 99.73 99.46 100.12 99.73 99.56
2 100.95 100.00 99.66 100.06 100.95 99.66 100.95 100.00 99.66 100.06
3 99.84 99.43 99.62 99.73 99.84 99.62 99.84 99.43 99.62 99.93
4 99.85 99.26 99.77 99.56 99.85 99.77 99.85 99.86 99.77 99.76
5 99.66 100.12 99.91 100.02 99.66 99.91 99.66 100.12 99.91 100.02
6 99.82 100.06 99.87 100.18 99.82 99.87 99.82 100.06 99.87 100.18
7 99.86 99.66 99.46 99.52 99.86 99.46 99.86 99.66 99.86 99.72
8 99.87 99.62 100.12 99.62 99.87 100.12 99.87 99.62 100.12 99.62
9 99.85 99.67 100.13 100.07 99.85 100.13 99.85 99.67 100.13 100.07
10 99.72 99.52 99.85 99.71 99.72 99.85 99.72 99.52 99.85 99.71
11 99.88 100.00 100.26 99.87 99.88 100.26 99.88 100.00 99.86 99.87
12 99.65 100.06 100.15 99.93 99.65 100.15 99.65 100.06 100.15 99.93
13 99.46 99.70 99.43 99.87 99.86 99.63 99.66 99.70 99.83 99.87
14 99.91 99.60 99.75 100.22 99.91 99.75 99.91 99.60 99.75 100.22
15 100.04 100.06 99.82 99.85 100.04 99.82 100.04 100.06 99.82 99.85
16 100.15 100.10 99.93 99.62 100.15 99.93 100.15 100.10 99.93 99.62
17 100.13 99.56 98.81 99.25 100.13 98.81 100.13 99.56 98.81 99.25
18 99.82 100.05 100.08 99.75 99.82 100.08 99.82 100.05 100.08 99.75
19 99.90 100.10 99.91 100.15 99.90 99.91 99.90 100.10 99.91 100.15
20 99.50 99.76 99.95 99.56 99.50 99.95 99.50 99.76 99.95 99.56
21 99.88 100.12 100.00 100.25 99.88 100.00 99.88 99.92 100.00 100.05
22 99.87 99.67 99.51 99.71 99.87 99.51 99.87 99.67 99.51 99.71
23 100.31 99.91 99.60 99.96 100.31 99.60 100.31 99.91 99.60 99.96
24 99.66 99.46 99.91 99.72 99.66 99.91 99.66 99.46 99.91 99.72
25 99.90 100.02 99.70 100.07 99.90 99.70 99.90 100.02 99.70 100.07

 Construct suitable control charts for variability as well as for average to infer whether the process of bottling is under statistical control or not. If it is out-of-control, also plot the revised control charts, if necessary.

A Mobility-as-a-Service (MaaS) provider company conducted a study to check the relationship of several variables with its weekly commuters. For this purpose, thirty cities were selected and the number of weekly commuters were recorded along with other variables like: the average petrol price (in ₹), population of the city, monthly income of commuters (in ₹), average parking rates per month (in ₹). The data are given in the following table:

City Number of Weekly Commuters Average Petrol Price Population of City (in ’000) Average Monthly Income of Commuters
(in ’00)		 Average Monthly Parking Rates
(in ₹)
1 17700 75 1900 580 1000
2 17540 75 1890 620 1000
3 17620 75 1880 640 1200
4 16260 77 1878 650 1200
5 16180 77 1850 655 1200
6 16340 77 1840 658 1400
7 16580 77 1825 820 1500
8 16020 82 1825 860 1500
9 15940 82 1820 880 1500
10 15892 82 1805 920 1600
11 15780 85 1810 963 1600
12 14820 95 1800 1057 1600
13 14660 95 1795 1133 1700
14 14660 96 1795 1160 2000
15 14580 96 1790 1180 2100
16 14420 96 1730 1183 2100
17 13380 97 1740 1265 2100
18 10070 120 1735 1300 2200
19 13220 102 1730 1325 2500
20 13540 102 1720 1380 2600
21 13700 102 1715 1401 3000
22 12100 107 1705 1450 3100
23 11124 113 1690 1500 3300
24 10900 125 1695 1520 3500
25 11108 114 1690 1560 3500
26 13668 104 1700 1600 3800
27 13780 90 1710 1620 4000
28 12108 118 1790 1590 3700
29 14668 108 1800 1630 4000
30 14780 94 1810 1650 4200

 Now determine the most appropriate regression model for the number of weekly commuters using stepwise approach at 5 % level of significance and interpret the results. Does the final regression model satisfy the linearity and normality assumptions?

An automobile manufacturing company examined the cars of a particular model to identify the number of defects during the final inspection stage. The total number of inspected carswere recorded for last 35 days along with the number of defects. The results are given in the following table:

Days Total Inspected Cars Number of Defects
1 45 6
2 35 2
3 40 1
4 30 2
5 40 5
6 25 1
7 35 3
8 30 4
9 30 6
10 40 2
11 35 11
12 35 1
13 25 3
14 40 7
15 45 2
16 40 6
17 30 5
18 35 2
19 30 7
20 45 3
21 28 1
22 38 4
23 33 11
24 33 1
25 43 4
26 38 2
27 38 5
28 28 4
29 43 6
30 48 1
31 43 3
32 33 4
33 38 3
34 33 2
35 48 5

Construct a suitable control chart for the number of defects to check whether the process is under statistical control or not. Also plot the revised control charts, if necessary.

A division of climatic change is interested in analysing the pattern of the CO2 concentrations in the air of a particular state in past years and then forecasting the CO2 concentrations for the upcoming years. The monthly mean CO2 concentrations ppm (parts per million) mixing ratio in dry air from January 2006 through December 2019 were recorded. The following monthly data are given in the following table for past 14 years:

Month 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019
January 299.16 299.62 300.60 301.67 303.71 305.00 306.77 308.32 309.90 310.41 312.87 313.88 315.04 316.33
February 299.94 300.40 301.60 302.17 304.23 305.63 307.26 309.41 310.70 311.68 313.59 314.62 315.70 316.82
March 300.41 300.87 302.57 303.86 305.54 306.93 309.10 310.69 311.70 312.04 314.11 316.23 316.38 318.29
April 301.72 302.18 303.72 305.07 306.79 307.95 309.88 311.51 312.65 314.27 316.07 316.62 318.38 319.91
May 302.13 302.92 303.68 305.82 306.95 308.05 310.47 312.02 313.28 314.92 316.38 317.53 318.93 319.41
June 301.09 302.43 303.17 305.12 307.00 308.27 310.31 311.54 312.42 314.40 315.78 316.87 318.26 318.74
July 300.10 300.85 301.96 303.81 305.37 306.64 308.41 309.88 311.02 313.16 313.96 315.00 316.44 316.92
August 298.14 299.01 299.80 301.56 303.47 304.68 306.74 307.75 308.97 311.11 311.71 312.86 314.55 315.03
September 296.36 297.51 297.98 300.30 301.46 302.77 305.07 306.05 306.84 309.11 309.86 311.55 313.21 313.69
October 296.29 297.41 297.57 300.22 301.29 303.09 305.07 306.13 307.00 309.15 310.13 311.57 312.67 313.15
November 297.23 298.25 299.20 301.37 302.76 304.29 306.22 307.45 308.20 310.38 311.84 313.11 314.09 314.57
December 298.55 299.97 300.58 302.49 303.80 305.76 307.38 308.85 309.63 312.02 313.32 314.49 315.27 315.75

a) Compute the seasonal indices using ratio to moving average method.
b) Obtain the deseasonalised values and then fit a linear trend line to the average annual CO2 concentrations using least squares method.
c) Convert the annual least-squared trend equation to a monthly trend equation.
d) Use the monthly trend equation and seasonal indices to forecast the CO2 concentrations for all twelve months of 2021.
e) Plot the original data, deseasonalised data and trend values.

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