X ~ N(70, 14). Suppose that you form random samples of 25 from this distribution. Let X be the random variable of averages. Let ?X be the random variable of sums. Find the 10th percentile. (Round your answer to two decimal places.)

The 10th percentile is 66.42.

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 normal distributions can be solved using the z-score formula.

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

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 and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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

X ~ N(70, 14).

This means that

Suppose that you form random samples of 25 from this distribution.

This means that

Find the 10th percentile.

This is X when Z has a pvalue of 0.1, so X when Z = -1.28.

By the Central Limit Theorem

The 10th percentile is 66.42.