Question

10. The population average for SAT score is 1020, with a population standard deviation of 100 a. What is the standard deviation of the distribution of , if many samples of size 64 is selected b. What is the probability that a random sample of 64 students will provide a sample mean SAT score within 10 points of the population mean?

Answer 1

10. $\small \bar{x}$ = 1020

$\small \sigma$ = 100

a) Standard deviation of the distribution of $\small \bar{x}$ if sampling size, n = 64 is given by,

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

= $\small 100/\sqrt{64}$

= 12.5

b) $\small \mu_{\bar{x}}$ = $\small \bar{x}$ = 1020

$\small \sigma_{\bar{x}}$ = 12.5

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

P(sample mean within 10 points of the population mean) = P(1010 < $\small \bar{x}$ < 1030)

= P(Z < (1030 - 1020)/12.5) - P(Z < (1010 - 1020)/12.5)

= P(Z < 0.8) - P(Z < -0.8)

= 0.7881 - 0.2119

= 0.5762