Question

In order to study the relationship between playing football and traumatic brain injuries, researchers placed accelerometers in the helmets of many high school football teams. Accelerometers measure the amount of hits that boys playing high school football absorb during a season. Suppose the average number of hits absorbed by a helmet during a season was 355 hits with a standard deviation of 80 hits. What is the probability on a randomly selected team of 48 players that the average number of head hits per player is between 340 and 360 hits

Answer 1

56.96% probability on a randomly selected team of 48 players that the average number of head hits per player is between 340 and 360 hits

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a single player:

`\mu = 355, \sigma = 80`

For the sample mean(of 48 playes).

`n = 48, s = (80)/(√(48)) = 11.547`

What is the probability on a randomly selected team of 48 players that the average number of head hits per player is between 340 and 360 hits?

This is the pvalue of Z when X = 360 subtracted by the pvalue of Z when X = 340. So

X = 360

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (360 - 355)/(11.547)`

`Z = 0.43`

`Z = 0.43` has a pvalue of 0.6664

X = 340

`Z = (X - \mu)/(s)`

`Z = (340 - 355)/(11.547)`

`Z = -1.30`

`Z = -1.30` has a pvalue of 0.0968

0.6664 - 0.0968 = 0.5696

56.96% probability on a randomly selected team of 48 players that the average number of head hits per player is between 340 and 360 hits