A magazine reports that the average bowling score for league bowlers in the Unit

Question

A magazine reports that the average bowling score for league bowlers in the United States is 157 with a standard deviation of 12, and that the scores are approximately normally distributed.

A) State the shape, the mean, and the standard deviation of the mean (xbar) of 15 bowlers selected at random. Justify each statement by showing your work or identifying the theorem that tells you your answer is correct.

B) What's the probabilty that a random sample of 15 bowlers would have an average score greater than 165?

C) What's the probability a random sample of 150 bowlers will have an average score greater than 165?

Explanation / Answer

Normal Distribution
Mean ( u ) =157
Standard Deviation ( sd )=12
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

a.
Mean ( u ) =157
Standard Deviation ( sd )=12/ Sqrt ( 15 ) =3.098
Number ( n ) = 15
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  

b.
P(X > 165) = (165-157)/12/ Sqrt ( 15 )
= 8/3.098= 2.582
= P ( Z >2.582) From Standard Normal Table
= 0.0049                  
                  

c.
P(X > 165) = (165-157)/12/ Sqrt ( 150 )
= 8/0.98= 8.165
= P ( Z >8.165) From Standard Normal Table
= 0                  

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