Question

Use the data in the table below to calculate Cp, Cpl, Cpu, and Cpk, assuming that the specifications are 3.75 ± 1.26. Do not round intermediate calculations. Round your answers to three decimal places. Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Observations 3.65 3.70 3.69 4.23 3.96 3.41 3.28 4.17 4.25 3.60 4.16 3.28 3.88 3.56 3.54 3.73 3.46 3.30 3.40 3.29 4.27 3.65 3.48 3.71 3.61 3.36 3.12 3.43 3.70 3.99 2.89 3.16 4.42 3.59 3.11 3.80 3.46 3.05 3.14 3.48 3.05 3.12 3.99 3.35 2.89 Sample 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Observations 4.15 3.11 2.89 3.66 2.80 3.63 3.85 3.81 4.07 3.93 3.85 4.04 3.83 3.42 4.30 3.60 3.16 3.71 3.68 3.32 3.58 3.57 4.99 3.78 4.44 3.66 3.48 4.50 3.99 3.68 3.20 3.11 3.09 3.98 3.28 4.34 3.63 3.81 3.82 3.55 3.87 3.28 3.59 4.12 2.98 Cp:   Cpl:   Cpu:   Cpk:   Construct the frequency distribution for the data. For calculating the cell width assume 8 cells, the lower limit of 2.49 inches and upper limit of 5.01 inches. If your answer is zero, enter "0". Round your answers for interval boundaries to three decimal places and round your answers for frequencies to the nearest whole number. Cell From To (inclusive) Frequency 0 1 2 3 4 5 6 7 8 Choose the correct histogram for the data. The correct histogram is _______ . A. $appletImage?dbid=572155105$ B. $appletImage?dbid=1791110289$ C. $appletImage?dbid=932667550$ D. $appletImage?dbid=2128063578$

Answer 1

Given details

Upper specification limit = 3.75

Lower specification limit = 1.26

Step 1:

Calculate Average(X-bar) and standard deviation for the given sample data.

• First calculate the average for each subgroup. Enter the "=Average(B3:AE3)" in AF3 cell and the copy the forumla till AF5 cell.
• To calculate the Average (X-bar), enter = "Average(AF3:AF5)" in any cell. Average (X-bar) = 3.63
• To calculate the standard deviation, enter = "STDEV(AF3:AF5)" in any cell. Standard deviation = 0.14

Table 1 - average of X-bar and standard deviation

5 | 6 | 7 | 8 | 9 | 10 | 11 12 13 14 15 16 17 18 | 19 | 20 | 21 | 22 | 23 | 24 25 26 27 | 28 | 29 | 30 Average 25 26 28 29 3.65 3.73 2.89 3.7 3.46 3.16 3.69 3.3 4.42 4.23 3.1 3.59 96 3.41 | 3.28 | 4.17 | 4.25 3,6 | 4.16| 3.28 .88| 3.56 3.54 4.15 3.11 12.: 3.66 2.8 | 3.63 3.85 | 3.81 | 4.07 9引3.55 4.04 3.83 | 3.42 4.3 | 3.72 3.29 4.273.65 3.48 3.713.61 3.36 3.12 3.33.73.993.6 3.16 3.713.683.323.58 3.57 4.993.78 4443.66 3.48453.99 3.68 3.69 3.11 .8 | 3.46 3.05 | 3.14 | 3.48 -.- .3 2.89 3.11 09 3.98 3.28 4.34 3.63 | 3.81 3.82 5: 87 | 3.28 | 3.59 4.12 2.98 3.47 Average (X-bar) 3.63SD 0.14

Step 2:

Calculate  C_p, C_pl, C_pu, and C_pk

_{Cp = (UJ SL _ LSL)/6 × standardde u iation}

  
_{(3.75-1.26)/6 0.14 3.045}

_{Cpl (Average(X-bar) _ LS L)|3 × standarddeu i ation}

_{(3.628-1.26)/3 × 0.14 0.107}

<sub>Cpu = (USL-Average(X-bar)/3 × standarddeviation


(3.75-3.628)/3 0.14 0.299</sub>

  

= Min ( 0.107,0.299) = 0.107

Step3:

Given lower limit = 2.49 and upper limit = 5.01

Steps to construct frequency distribution table

• Data range for frequency calculation is B3:AE5
• Enter the class intervals in the cell range C11 to D19
• Enter the formula in cell H for frequency distribution
• Cell E11: =COUNTIFS($B$3:$AE$5,">0",$B$3:$AE$5,"<=2.49")
• Cell E12: =COUNTIFS($B$3:$AE$5,">2.49",$B$3:$AE$5,"<=2.80")
• Cell E13: =COUNTIFS($B$3:$AE$5,">2.80",$B$3:$AE$5,"<=3.12")
• Cell E14: =COUNTIFS($B$3:$AE$5,">3.12",$B$3:$AE$5,"<=3.43")
• Cell E15: =COUNTIFS($B$3:$AE$5,">3.43",$B$3:$AE$5,"<=3.75")
• Cell E16: =COUNTIFS($B$3:$AE$5,">3.75",$B$3:$AE$5,"<=4.06")
• Cell E17: =COUNTIFS($B$3:$AE$5,">4.06",$B$3:$AE$5,"<=4.38")
• Cell E18: =COUNTIFS($B$3:$AE$5,">4.38",$B$3:$AE$5,"<=4.69")
• Cell E19: =COUNTIFS($B$3:$AE$5,">4.69",$B$3:$AE$5,"<=5.01")

The result of frequency table and relevant graph is shown below

Table 2 - frequency table

| | MIN AA AB AD AE 9 101112 13 14 15 16 1718 1920 2122232425 26 27 28 29 30Average 3.65 3.73 2.89 3.7 3.46 3.16 3.69 3.3 4.42 4.23 3.4 3.59 3.72 3.69 3.47 3.9 3. 4.1 4.2 3. 4.1 :.2 3.& 3.5б 3.54 | 4.15 3.11 2.89 3.66 2. 3. . . 4. , , 4.04 3.83 3.42 4.3 3.2 4.2 .6 4 · . I 3.43 3.7 3.99 3.6 | 3.16 3.71 3.68 . .5 4. .: 4.44 . 3.48 4.5 3.99 3.68 Average (X-bar) 3.63SD 0.14 Frequency Table Upper Lowe Limit limit 2.49 2.8 3.12 3.43 3.75 4.06 4.38 4.69 5.01 10 35 30 2.49 2.8 3.12 3.43 3.75 4,06 4.38 4.69 12 12 17 15 15 16 17 18 19 20 10 2.49 2.8 3.12 3.43 3.75 4.06 4.38 4.69 5.01

Therefore, the correct answer is D.