In a recent study of longterm effects of concussions in foot

In a recent study of long-term effects of concussions in football players, researchers at Virginia Tech concluded that college football players receive a mean of 50 strong blows to head that average 40G (40 times the force of gravity). Assume the standard deviation of the number of blows to the head is 16. Suppose you select a simple random sample of n=60 college football players and install sensors on their helmets that records x ,the number of strong blows to the head. Note that since x is a count, the distribution of x cannot be modeled by a normal distribution.

(a). Let Xbar denote the sample mean of the number of strong blows to the head received by the 60 football players. Find the following

(i). mean(Xbar)

(ii). SD(Xbar)

(b). What is the sampling distribution model for (Xbar )? What important theorem in statistics do you support your claim with?

(c). What is the probability that in a sample of 60 football players the mean number of strong blows to the head is 45 or less? Would this be unusual?

Solution

(a)

i)

Mean(Xbar) = 50 Answer

ii)

SD(Xbar) = 16 / sqrt(60) = 2.066 Answer

(b)

The sample distribution is the distribution resulting from the collection of actual data. A major characteristic of a sample is that it contains a finite (countable) number of scores, the number of scores represented by the letter N.

Theoram is Central Limit Theoram.

(c)

P(X<=45) = P( z <= (45-50)/2.066)

= P( z <= -2.42)

= 0.0078 Answer

It is unusual.