Chapter 13 021.1-Q21.5 About A% of values drawn from a normal distribution are w
ID: 380493 • Letter: C
Question
Chapter 13 021.1-Q21.5 About A% of values drawn from a normal distribution are within one standard deviation away from the mean; about B% of the values lie within two standard deviations, and about C% are within three standard deviations. This fact is known as the A-B-C (empirical) rule. or the 3-sigma rule. What are the percentages, rounded to nearest whole number ie. 1%, 2%, etc. 2. B 3 standard deviations of the mean 2 standard deviations 1 standard deviation Continued in next page.... Page 8 of 12Explanation / Answer
1). For A, x ranges from ( - ) to ( + )
z = (x - )/. Therefore, z ranges from ( - - )/ to ( + - )/ or -1 to +1
Cumulative Distribution Function, F(z) ranges from F(-1) =NORMSDIST(-1) = 0.1587 to F(1) =NORMSDIST(1) =0.8413
Therefore Area under curve (Percentage) is :
A = 0.8413 - 0.1587 = 0.6827 ~ 68 %
______________
2). For B, x ranges from ( - 2) to ( + 2)
z = (x - )/. Therefore, z ranges from ( - 2 - )/ to ( + 2 - )/ or -2 to +2
Cumulative Distribution Function, F(z) ranges from F(-2) =NORMSDIST(-2) = 0.0228 to F(2) =NORMSDIST(2) =0.9772
Therefore Area under curve (Percentage) is :
B = 0.9772 - 0.0228 = 0.9545 ~ 95 %
______________
3). For C, x ranges from ( - 3) to ( + 3)
z = (x - )/. Therefore, z ranges from ( - 3 - )/ to ( + 3 - )/ or -3 to +3
Cumulative Distribution Function, F(z) ranges from F(-3) =NORMSDIST(-3) = 0.0013 to F(3) =NORMSDIST(3) =0.9987
Therefore Area under curve (Percentage) is :
C = 0.9987 - 0.0013 = 0.9973 ~ 100 %
4) a.
USL = 170
LSL = 150
= 160
= 2.5
Cpk = MIN[(USL-)/3, (-LSL)/3]
= MIN[(170-160)/(3*2.5), (160-150)/(3*2.5)]
= MIN[1.33, 1.33]
= 1.33
b.
USL = 170
LSL = 150
= 162.5
= 2.5
Cpk = MIN[(USL-)/3, (-LSL)/3]
= MIN[(170-162.5)/(3*2.5), (162.5-150)/(3*2.5)]
= MIN[1.00, 1.67]
= 1.00
5) a. Cpk is equal to 1.33, which is the threshold for process capability. Therefore, process is capable.
b. Cpk is less than 1.33, which is below the threshold for process capability. Therefore, process is NOT capable.
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