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Question: Women constitute 24% of the science and engineering workforce. In a random sample of 500 science and engineering workforce, what is the probability that more than 130 are women?


A ) 13.57%
B ) 63.57%
C ) 86.43%
D ) 86.34%

Steps to derive

1 The probability of a woman workforce p = 24100,q = 1 - p = 76100.
Sample size, n = 500.

2 n·p = 500 × 24100 = 120 > 5
n·q = 500 × 76100 = 380 > 5
So, the use of normal approximation to the binomial distribution is permissible.

3 Mean, μ = n·p = 500 × 24100 = 120

4 Standard Deviation, σ = npq = 500×24100×76100 = 9.55

5 Write the problem in probability notation: P (x > 130), where x denotes the number of women.

6 P (x > 130 + 0.5) = P (x > 130.5)
 [Rewrite the problem using the continuity correction factors.]

7 The corresponding area under the normal curve distribution is shown in the diagram.


8 The z value corresponding to x = 130.5 is given by, z = (x-μ)σ = (130.5-120)9.55 = 10.59.55 = 1.1

9 The area between z = 0 and z = 1.10 is 0.3643 by referring to the table.
So, the area for z > 1.10 is 0.5 - 0.3643 = 0.1357.

10 So, the probability that more than 130 are women is 13.57 %.



Hence the right answer is Option A

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