Question

A large retirement community has a current population more than 5,000 residents. The distribution of ages of all residents is left skewed with a mean of 65.5 years and a standard deviation of 12.5 year. Suppose you are to conduct a survey and take a random sample of 100 residents of the community. Which of the following corretly describe how to find the probability that you obtain a sample mean age that is younger than 64 years? a) Find the area to the left of z = -0.12 under a standard normal curve. b) Find the area to the right of z = -1.2 under a standard normal curve. c) Find the area to the left of z = -1.2 under a standard normal curve. d) Find the area to the right of z = -0.12 under a standard normal curve. e) None of above

Answer 1

c) Find the area to the left of z = -1.2 under a standard normal curve.

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probabiliy distribution

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, or the area to the left of Z in the normal curve. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, which is the area to the right of Z in the normal curve.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 65.6, \sigma = 12.5, n = 100, s = (12.5)/(√(100)) = 1.25`

Which of the following corretly describe how to find the probability that you obtain a sample mean age that is younger than 64 years?

Area to the left of z when X = 64 under the standard normal curve. We have to find Z.

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

Applying the Central Limit Theorem

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

`Z = (64 - 65.6)/(1.25)`

`Z = -1.2`

So the correct answer is:

c) Find the area to the left of z = -1.2 under a standard normal curve.