8 According to the most recent Labor Department data 105 of

8. According to the most recent Labor Department data, 10.5% of engineers (electrical, mechanical, civil, and industrial) were women. Suppose a random sample of 50 engineers is selected. a. If the random sample of 50 engineers contained 8 women, what is the sample proportion of women? Provide the correct notation, and value. b. (4pts) How likely is it that the random sample of 50 engineers will contain 8 or more women in these positions? (Give the proper probability statements/notation, show work, and give value to 4 decimal places) c. (4pts) How likely is it that the random sample will contain fewer than 5 women in these positions? (Give the proper probability statements/notation, show work, and give value to 4 decimal places) d. (2pts) If the random sample included 200 engineers, how would this change your answer to part b? Be as specific as possible.

Solution

a)
Proportion rate ( P )= x/n = 8/50 = 6.25

b)
Mean ( np ) = 50 * 10.5 = 0.21
Standard Deviation ( npq )= 2*0.105*0.895 = 0.4335
Normal Distribution = Z= X- u / sd                   

P(X > 8) = (8-5.25)/2.1677
= 2.75/2.1677 = 1.2686
= P ( Z >1.269) From Standard Normal Table
= 0.1023                  

c)
P(X < 5) = (5-5.25)/2.1677
= -0.25/2.1677= -0.1153
= P ( Z <-0.1153) From Standard NOrmal Table
= 0.4541                  
d)
For n=200
Mean ( np ) = 200 * 10.5 = 21
Standard Deviation ( npq )= 200*0.105*0.895 = 4.3353
P(X > 8) = (8-21)/4.3353
= -13/4.3353 = -2.9986
= P ( Z >-2.999) From Standard Normal Table
= 0.9986