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 the sample has exactly 130 women workforce?
A ) 4.23% B ) 3.24% C ) 3.42% D ) 2.54%
Steps to derive
1 The probability of a woman workforce p = 24100 q = 1 - p = 76100 Sample size, n = 500
2n·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 = n⋅p⋅q = 500×24100×76100 = 9.55
5 Write the problem in probability notation: P (x = 130), where x denotes the number of women.
6 P (130 - 0.5 < x < 130 + 0.5) i.e. P (129.5 < 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 z1 value corresponding to x = 129.5 is given by, z = (x-μ)σ = (129.5-120)9.55 = 9.59.55 = 0.99
9 The z2 value corresponding to x = 130.5 is given by, z = (x-μ)σ = (130.5-120)9.55 = 10.59.55 = 1.10
10 The area between z = 0 and z = 0.99 is 0.3389 and the area between z = 0 and z = 1.10 is 0.3643 by referring to the table.
11 Subtracting the area to get the approximate value is 0.3643 - 0.3389 = 0.0254
12 So, the probability that there will exactly be 130 women workforce in the sample is 2.54%.
