Question

"The percent of fat calories that a person consumes each day is normally distributed with a mean of 36 and a standard deviation of 10. Suppose that 25 individuals are randomly chosen. Find the first quartile for the average percent of fat calories."

Answer 1

The first quartile for the average percent of fat calories is 34.65.

Explanation:

To solve this question, we use 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 = (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 random variable X, with mean
`\mu` and standard deviation
`\sigma`, a large sample size can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 36, \sigma = 10, n = 25, s = (10)/(√(25)) = 2`

Find the first quartile for the average percent of fat calories.

This is the value of X when Z has a pvalue of 0.25. So X when Z = -0.675.

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

By the Central limit theorem

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

`-0.675 = (X - 36)/(2)`

`X - 36 = -0.675*2`

`X = 34.65`

The first quartile for the average percent of fat calories is 34.65.