Question

A common test of strength engaged in by high school students at Elmhurst HS is seeing how much weight they can lift from the floor to a standing position (called a "deadlift"). Assume the weight lifted in such a manner is normally distributed and has a mean of 225 lbs. with a standard deviation of 50 lbs.
(a) Find the probability that 1 randomly selected male student has the best lift less than 200 lbs.
(b) If a sample of 25 students is tested, find the probability the sample mean will be over 245 lbs.
(c) Why can the normal distribution be used in part b even though the sample size is < 30?

Answer 1

Answer:

a) 30.85% probability that 1 randomly selected male student has the best lift less than 200 lbs.

b) 2.28% probability the sample mean will be over 245 lbs.

c) Because the underlying population(weight the students can lift) is normally distributed.

Step-by-step explanation:

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

Normal probability distribution

Problems of normally distributed samples are solved using 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.

In this problem, we have that:

$\mu = 225, \sigma = 50$

(a) Find the probability that 1 randomly selected male student has the best lift less than 200 lbs.

This is the pvalue of Z when X = 200. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{200 - 225}{50}$

$Z = -0.5$

$Z = -0.5$ has a pvalue of 0.3085

30.85% probability that 1 randomly selected male student has the best lift less than 200 lbs.

(b) If a sample of 25 students is tested, find the probability the sample mean will be over 245 lbs.

Now $n = 25, s = \frac{50}{\sqrt{25}} = 10$

This probability is 1 subtracted by the pvalue of Z when X = 245. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

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

$Z = \frac{245 - 225}{10}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

1 - 0.9772 = 0.0228

2.28% probability the sample mean will be over 245 lbs.

(c) Why can the normal distribution be used in part b even though the sample size is < 30?

The sample size being at least 30 condition is only if the underlying population is not normally distributed. In this case, it is, so we use the normal distribution in part b.