Question

10. (1 pt) 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 Xbe 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

Here, the population mean, $\mu$ = 2100 sqft

and population standard deviation, $\sigma$ = 500 sqft

For sampling distribution of mean of sample size n = 625,

Mean, $\mu_{\bar{x}}$ = $\mu$ = 2100 sqft

Standard error $\sigma_{\bar{x}}$ = $\sigma/\sqrt{n}$

= $500/\sqrt{625}$

= 20

P($\bar{x}$ < A) = P(Z < ($\bar{x}$ - A)/$\sigma_{\bar{x}}$)

P(mean living space is less than 2069) = P($\bar{x}$ < 2069)

= P(Z < (2069 - 2100)/20)

= P(Z < -1.55)

= 0.0606 (from standard normal distribution table)