Question The lengths, in inches, of adult corn snakes have an unknown distribution with mean 51 and standard deviation 9 inches. A

sample, with size n = 43, is randomly drawn from the population and the mean is taken. What is the probability that the

mean is less than 50.2 inches?

0.281 = 28.1% probability that the mean is less than 50.2 inches.

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.

Mean 51 and standard deviation 9 inches.

This means that

A sample, with size n = 43

By the Central Limit Theorem,

What is the probability that the mean is less than 50.2 inches?

This is the pvalue of Z when X = 50.2. So

By the Central Limit Theorem

has a pvalue of 0.281

0.281 = 28.1% probability that the mean is less than 50.2 inches.