SOLUTION: Men's heights are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches, while women's heights are normally distributed with a mean of 63.6 inches

(a)  Every normal distribution curve is a bell shaped curve.

     In this problem, they want you determine the area under this specific normal curve
     on the right from the raw mark of 72 inches.  This area is the sought probability.


     Go to web-site https://onlinestatbook.com/2/calculators/normal_dist.html
     and find there free of charge online calculator, specially developed for this purpose.

     Input the mean value 69.0 and the standard deviation value 2.8;
     input 72 in the window "Above"; then click "Recalculate".

     You will get the ANSWER  0.142  in the window "Probability".

     The auxiliary plot will show you the area of the interest.


     Now, when you know "what to do", I will tell you that you can get the same result 
     using your calculator  TI-83 or TI-84 (or whatever).

     For it, use the calculator' standard function normalcdf with parameters

                       z1  z2  mean SD   

         P = normalcdf(72,9999,69.0,2.8) = 0.142.

     You will get the same value of the probability 0.142 from your calculator.

     At this point, the solution to part (a) is completed.



(b)  For question (b), do THE SAME, changing your input numbers accordingly.



(c)  In part (c), they want you determine the raw z-score in such a way, that the area 
     under the normal curve on the left of this z-score would be 95% = 0.95.


     To do it, go to ANOTHER website https://onlinestatbook.com/2/calculators/inverse_normal_dist.html
     and find there ANOTHER free of charge online calculator, specially developed for this purpose.

     Input the mean value 0.95 in the window "Area"; input 69.0 in the window "Mean";
     input 2.8 in the window "SD" (standard deviation); then click "Recalculate".

     Click for the window "Below" as you want the area below 0.95;
     and finally click the button "Recalculate".

     You will get the ANSWER  73.6056, which represent the sought-for/(desired) z-score.
     (the height of the door in inches in this problem).

     The auxiliary plot will show you the area of the interest.


     Now, when you know "what to do", I will tell you that you can get the same result 
     using your calculator  TI-83 or TI-84 (or whatever).

     For it, use the calculator' standard function invNorm with parameters

                     area mean SD   

         P = invNorm(0.95,69.0,2.8) = 73.0056.

     You will get the same value 73.0056 for the z-score, which is the height of the door in inches 
     in this problem.

     The auxiliary plot will show you the area of the interest.


     At this point, the solution to part (c) is completed.