During the summer of 2014, Coldstream Country Club in Cincinnati, Ohio collected data on 443 rounds of golf played from its white tees. The data for each golfer's score on the twelfth hole are below:

Hole 12 Score:
6,7,5,7,5,5,5,4,5,5,5,5,5,4,5,5,5,7,5,7,7,5,6,5,5,5,6,5,6,5,6,5,5,5,6,5,5,6,5,6,5,5,6,5,6,6,6,4,3,4,6,8,6,6,6,4,6,5,6,5,6,6,6,4,5,5,5,5,7,5,6,5,6,5,4,6,5,5,5,5,5,7,5,5,4,5,5,6,6,5,5,6,5,5,4,6,5,5,5,5,5,6,5,5,4,6,5,5,5,6,5,5,4,6,7,5,5,5,6,4,6,4,5,7,6,6,4,5,4,5,6,6,5,5,5,6,5,5,5,6,4,4,5,5,4,5,8,5,4,5,6,5,4,6,6,5,4,5,6,5,5,4,6,6,6,4,5,6,6,6,6,5,5,6,6,6,6,6,6,7,5,6,7,5,6,6,5,6,6,5,6,5,6,7,6,6,5,5,5,5,6,5,4,8,6,5,5,5,5,6,7,6,7,6,6,4,5,6,4,5,5,7,5,5,5,5,5,7,5,6,6,5,4,5,5,5,5,4,6,5,5,5,5,6,5,5,6,5,7,5,5,5,5,5,5,5,4,5,5,6,5,4,5,5,5,6,5,5,6,6,6,6,6,7,6,6,5,6,5,4,6,5,5,5,5,6,6,5,7,6,5,6,5,5,8,5,3,6,6,6,6,5,3,6,7,5,5,4,6,5,6,5,5,4,5,5,7,5,5,7,5,5,6,6,6,6,6,5,54,5,6,5,6,5,6,5,6,5,5,5,5,5,5,4,5,5,4,5,6,6,6,5,6,4,6,5,6,4,4,3,5,4,5,6,5,5,6,4,6,4,4,5,5,6,7,5,4,7,5,6,5,5,4,5,6,5,5,4,5,5,6,5,6,7,6,4,5,6,5,4,4,4,5,5,4,4,6,4,5,5,7,5,5,6,4,5,6,5,5,6,6,5,5,6,7,5,6,5,5,5,4,4,6,5,4,6,6,6,6,5

(Round your answers to four decimal places.)
(a) Construct an empirical discrete probability distribution for the player scores on the twelfth hole.
(b) A par is the score that a good golfer is expected to get for the hole. for hole number 12, par is four. What is the probability of a player scoring less than or equal to par on hole number 12?
(c) What is the expected score for hole number 12?
(d) What is the variance for hole number 12?
(e) What is the standard deviation for hole number 12?

[tex]\left|\begin{array}{c|c|c|c}---&----&----&---\\Score(x)&Frequency,F(x)&P(x)&P(X\leqx) \\----&----&-----&-----\\3&4&P(X=3)=\frac{4}{443}&P(X\leq 3) =\frac{4}{443}\\4&57&P(X=4)=\frac{57}{443}&P(X\leq 4) =\frac{61}{443}\\5&212&P(X=5)=\frac{212}{443}&P(X\leq 5) =\frac{269}{443}\\6&139&P(X=6)=\frac{139}{443}&P(X\leq 6) =\frac{336}{443}\\7&27&P(X=7)=\frac{27}{443}&P(X\leq 7) =\frac{393}{443}\\8&4&P(X=8)=\frac{4}{443}&P(X\leq 3) =\frac{443}{443}\\----&----&----&----\\&443\\----&---&----&----\end{array}\right|\\[/tex]

(b)The probability of a player scoring less than or equal(4) to par on hole number 12.

[tex]P(X\leq 4) =\frac{61}{443}[/tex]

(c)Expected Score for Hole Number 12

[tex]\left|\begin{array}{c|c|c}-----&----&---&\\Score(x)&P(x)&x.P(x) \\----&----&-----\\3&P(X=3)=\frac{4}{443}&0.0271\\4&P(X=4)=\frac{57}{443}&0.5147\\5&P(X=5)=\frac{212}{443}&2.3928\\6&P(X=6)=\frac{139}{443}&1.8826\\7&P(X=7)=\frac{27}{443}&0.4266\\8&P(X=8)=\frac{4}{443}&0.0722\\----&----&----\\&443&5.3160\\----&----&----\end{array}\right|\\[/tex]

Expected value of x, E(X)=5.3160

(d)Variance

[tex]E(X-\mu)^2=E(X^2-2X\mu-\mu^2)=E(X^2)-2\muE(X)+\mu^2\\=E(X^2)-\mu^2\\=\sum x^2P(x)-\mu^2[/tex]

[tex]\left|\begin{array}{c|c|c}-----&----&---&\\Score(x)&P(x)&x^2.P(x) \\----&----&-----\\3&P(X=3)=\frac{4}{443}&0.0813\\4&P(X=4)=\frac{57}{443}&2.0587\\5&P(X=5)=\frac{212}{443}&11.9639\\6&P(X=6)=\frac{139}{443}&11.2957\\7&P(X=7)=\frac{27}{443}&2.9865\\8&P(X=8)=\frac{4}{443}&0.5779\\----&----&----\\&&28.9639\\----&----&----\end{array}\right|\\[/tex]

Variance=[tex]28.9639-5.3160^2=0.7037[/tex]

(e)Standard Deviation

[tex]S.D.=\sqrt{Variance}=\sqrt{0.7037} =0.8389[/tex]