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

(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 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:

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

By the Central Limit Theorem

has a pvalue of 0.0606.

So the correct answer is:

(d) .0606

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Step-by-step explanation:

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