Question

The living space of all homes in a city has a mean of 2100 square feet and a standard deviation of 500 square feet. Let be the mean living space for a random sample of 625 homes from this city. The probability that this mean living space is less than 2069 square feet is___________. (a) .0478(b) .9394(c) .8788(d) .0606(e) .5239

Answer 1

Answer:

(d) .0606

Step-by-step explanation:

To solve this question, we have 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 $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 2100, \sigma = 500, n = 625, s = \frac{500}{\sqrt{625}} = 20$

The probability that this mean living space is less than 2069 square feet is:

This probability is the pvalue of Z when X = 2069. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{2069 - 2100}{20}$

$Z = -1.55$

$Z = -1.55$ has a pvalue of 0.0606.

So the correct answer is:

(d) .0606