"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."

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

Step-by-step 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 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 random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

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.

By the Central limit theorem

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

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step-by-step explanation:

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