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likoan [24]
2 years ago
9

In order to study the relationship between playing football and traumatic brain injuries, researchers placed accelerometers in t

he 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
Mathematics
1 answer:
disa [49]2 years ago
4 0

Answer:

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

Step-by-step 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 = \frac{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 = \frac{\sigma}{\sqrt{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 = \frac{80}{\sqrt{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 = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{360 - 355}{11.547}

Z = 0.43

Z = 0.43 has a pvalue of 0.6664

X = 340

Z = \frac{X - \mu}{s}

Z = \frac{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

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