2017 HMK 3 3. Ten pieces a below as length in inches: re sampled from a manufact

Question

2017 HMK 3 3. Ten pieces a below as length in inches: re sampled from a manufacturing process and the resulting measurements are shown 0.851 0.848 0.853 0.855 0.856 0.857 0.845 0.851 0.855 0.849 a- Calculate Descriptive Stats: Mode, Median, Mean, Range, STDEV. b- The print specification is 0.850 +/-0.005. Chart the associated normal distribution, and apply vertical lines at the print tolerance values. c- What is Probability next part taken will be s 0.851? d-What is Probability next part taken will be 2 0.854? e-What is Probability that next part taken will be between 0.853 and 0.854 written as: P = 0.8535 x 0.854 = 96 f- What is the probability that the next part taken will be outside of specification?

Explanation / Answer

The measurements are given as below:
0.851, 0.848, 0.853, 0.855, 0.856, 0.857, 0.845, 0.851, 0.855, 0.849

Mode is 0.851 and 0.855, as both occur twice.

Median = For checking median we first need to sort the terms in order.
Sorting give the below order of terms:
0.845 0.848 0.849 0.851 0.851 0.853 0.855 0.855 0.856 0.857
Median will be the mean of 5th and 6th term.
So median = (0.851 + 0.853)/2 = 0.852

Mean = (0.851 + 0.848 + 0.853 + 0.855 + 0.856 + 0.857 + 0.845 + 0.851 + 0.855 + 0.849)/10 = 0.852

Range is 0.845 to 0.857

First we can calculate the variance
Variance = (Sum of squares)/N-1

((0.851 - 0.852)^2 + (0.848 - 0.852)^2 + (0.853 - 0.852)^2 + (0.855 - 0.852)^2 + (0.856 - 0.852)^2 + (0.857 - 0.852)^2 + ( 0.845 - 0.852)^2 + (0.851- 0.852)^2 + (0.855- 0.852)^2 + (0.849- 0.852)^2) /(10 - 1)

= 1.51 * 10^-5

Standard deviation = sqrt(Variance) = 0.003887301

c) Probability that next part will be <=0.851 is:
z score corresponding to 0.851 is (0.851 - 0.852)/0.003887301 = -0.2572479
P(x <= 0.851) = P(z <= -0.2572479) = 0.39849

d) Probability that next part will be >=0.854 is:
z score corresponding to 0.854 is (0.854 - 0.852)/0.003887301 = 0.5144958
P(x >= 0.854) = 1 - P(x < 0.854) = 1 - P(z < 0.5144958) = 1 - 0.69655 = 0.30345

e) Probability that next part will be between 0.853 and 0.854 is:

P(0.853 <= X <= 0.854) = P(X <= 0.854) - P(X <= 0.853)
Z score corresponsing to 0.853 = (0.853 - 0.852)/0.003887301 = 0.2572479
Z score corresponsing to 0.854 = (0.854 - 0.852)/0.003887301 = 0.5144958

P(0.853 <= X <= 0.854) = P(X <= 0.854) - P(X <= 0.853) = P(z <= 0.5144958) - P(z < 0.2572479)
P(0.853 <= X <= 0.854) = 0.69655 - 0.60151 = 0.09504

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