According to the most recent Labor Department data 105 of en

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. How likely is it that the random sample of 50 engineers will contain 8 or more women in these postitions? (Give the proper probablity statements/notation, show work, and give value to 4 decimal places)

C. How likely is it that the random sample will contain fewer than 5 women in these positions? (Give the proper probablity statements/notation, show work, and give value to 4 decimal places)

D. If the random sample included 200 engineers, how would this change your answer in part b? be specific as possible.

Solution

a)

p^ = 8/50 or

p^ = 0.16 [ANSWER, where p^ denotes sample proportion]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    50      
p = the probability of a success =    0.105      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   7   ) =    0.850669594
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.149330406 [ANSWER]

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c)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    50      
p = the probability of a success =    0.105      
x = the maximum number of successes =    5      
          
Then the cumulative probability is          
          
P(at most   5   ) =    0.569867893 [ANSWER]

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d)

If we have 200 engineers instead, then part b will contain 32 women instead.

Then, the probability will be smaller:

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    200      
p = the probability of a success =    0.105      
x = our critical value of successes =    32      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   31   ) =    0.98933523
          
Thus, the probability of at least   32   successes is  
          
P(at least   32   ) =    0.01066477

As we see, the probability became smaller than part b.