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Twenty percent of the employees of a large company are female. Use the normal approximation
of the binomial probabilities to answer the following questions. What is the probability
that in a random sample of 80 employees
(a) exactly 16 will be female?
(b) 14 or more will be female?
(c) 15 or fewer will be female?
(d) 18 or more will be female
(e) exactly 17 will be female?
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You have a binomial distribution with large number of trials n = 80
and individual probability of success p = 20% = 0.2.
You want to approximate it by the normal distribution.
You should use the mean value m = p*n = 0.2*80 = 16 and standard deviation
SD = = = 3.57771.
Also, you should use the continuity correction factor.
About approximation of the binomial distribution by normal distribution and continuity correction factor
see your textbook and/or these Internet sources
https://www.statology.org/normal-approximation
https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/normal-approximation-to-the-binomial/
https://online.stat.psu.edu/stat414/lesson/28/28.1
https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/06%3A_Continuous_Random_Variables_and_the_Normal_Distribution/6.04%3A_Normal_Approximation_to_the_Binomial_Distribution
For calculations, you may use your calculator (function normcdf), or Excel spreadsheet (function NORMDIST);
or online calculator https://onlinestatbook.com/2/calculators/normal_dist.html
(a) In this case, you should find the area under the normal curve between z-scores 15.5 and 16.5
(using the correction factor)
z1 z2 mean SD <<<---=== formatting pattern
P(x = 16) = normcdf(15.5, 16.5, 16, 3.57771) = 0.1111 (rounded).
(b) In this case, you should find the area under the normal curve on the right from 13.5
(using the correction factor)
z1 z2 mean SD <<<---=== formatting pattern
P(x >= 14) = normcdf(13.5, 9999, 16 3.57771) = 0.7577 (rounded).
(c) In this case, you should calculate the area under the normal curve on the left of the mark z = 15.5
(using the correction factor)
z1 z2 mean SD <<<---=== formatting pattern
P(x <= 15) = normcdf(-9999, 15.5, 16, 3.57771) = 0.4444 (rounded).
(d) Do it as (b).
(e) Do it as (a).
Solved.
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If you are a beginner student in learning probability distributions, I advise you to start learning
this kind of computations using the online calculator, to which I referred above.
It provides a graphical support, so at each step you do understand what you are doing.
In addition, this graphical support prevents you from making mistakes.
When you learn enough this kind of computations, you can switch to your regular calculator,
but even then you may use the online calculator for checking purposes.