Question

4. Assume that the mean weight of all NFL players is 245.7 pounds, with a standard deviation of 34.5 pounds. A random sample of 32 NFL players is selected for study.

A) What is the Shape, mean (expected value) and standard deviation of the sampling distribution of the sample mean for this study?

B) For the sample of 32 players, what is the probability that the sample mean weight is greater than 240 pounds?

C) What is the probability that an individual player from the NFL weighs more than 240 pounds?

D) For the sample of 32, what is the probability that the sample mean weight is between 240 and 250 pounds?

Answer 1

Let X = weight of all NFL players

It is given that the mean weight of all NFL players = = 245.7

and standard deviation of weight of all NFL players = = 34.5

n = sample size = 32 ( which is sufficiently large > 30 ) for applying Central Limit Theorem.

A) For large n ( > 30 ) the sampling distribution of sample mean follows approximately normal distribution with mean and standard deviation are as follows:

Mean of sample mean = = 245.7

Standard deviation of sample mean ( ) is as follows:

B) For n = 32 , we want to find the probability that the sample mean weight is greater than 240 pounds

Mathematically,

Lets find P( < 240) using excel:

  P( < 240) = "=NORMDIST(240,245.7,6.0988,1)" = 0.1750

Plug this value in equation ( 1 ), we get,

C) What is the probability that an individual player from the NFL weighs more than 240 pounds?

For individual we use sttandard deviation = 34.5

P( X > 240 ) = 1 - P( X < 240 ) ....( 2 )

P( X < 240 ) = "=NORMDIST(240,245.7,34.5,1)" = 0.4344

Plug this value in equation ( 2 ) , we get

P( X > 240 ) = 1 - 0.4344 = 0.5656

D) For the sample of 32, what is the probability that the sample mean weight is between 240 and 250 pounds?

That is we want to find ,

P( < 250 ) = "=NORMDIST(250,245.7,6.0988,1)" = 0.7596

And from part a )  P( < 240 ) = 0.1750

Plug these values in equation ( 3 ) , we get