A study investigating the relationship between age and annual medical expenses randomly samples 400 individuals in a city. It is hoped that the sample will have a similar mean age as the entire population. Complete parts a and b below.

a. If the standard deviation of the ages of all individuals in the city is σ=16,find the probability that the mean age of the individuals sampled is within one year of the mean age for all individuals in the city. (Hint: Find the sampling distribution of the sample mean age and use the central limit theorem. You don't have to know the population mean to answerthis, but if it makes it easier, use a value such as μ=30.)

The probability is

(Round to three decimal places as needed.)

The probability is 0.789

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 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.

In this problem, we have that:

Find the probability that the mean age of the individuals sampled is within one year of the mean age for all individuals in the city.

Using , this is the pvalue of Z when X = 31 subtracted by the pvalue of Z when X = 29. So

X = 31

By the Central Limit Theorem

has a pvalue of 0.8944

X = 29

has a pvalue of 0.1056

0.8944 - 0.1056 = 0.7888

Rounded to three decimal places, 0.789

The probability is 0.789

step-by-step explanation:

c

step-by-step explanation: