Question

8. Assume that SAT scores are normally distributed with mean 1518 and standard deviation 325. ROUND YOUR ANSWERS TO 4 DECIMAL PLACES

a. If 100 SAT scores are randomly selected, find the probability that they have a mean less than 1500.___________

b. If 64 SAT scores are randomly selected, find the probability that they have a mean greater than 1600

c. If 25 SAT scores are randomly selected, find the probability that they have a mean between 1550 and 1575

d. If 16 SAT scores are randomly selected, find the probability that they have a mean between 1440 and 1480.

e. In part c and part d, why can the central limit theorem be used even though the sample size does not exceed 30? ____________________________________________________________________

Answer 1

8)

a)

$\sigma$ _{$\bar x$} = $\sigma$ / $\sqrt$ n = 325 / $\sqrt$ 100 = 32.5

P( $\bar x$ < 1500) = P(( $\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (1500 - 1518) / 32.5)

= P(z < -0.554)

= 0.2898

(b)

$\sigma$ _{$\bar x$} = $\sigma$ / $\sqrt$ n = 325 / $\sqrt$ 64 =  40.625

P( $\bar x$ > 1600) = 1 - P( $\bar x$ < 1600)

= 1 - P[( $\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (1600 - 1518) / 40.625]

= 1 - P(z < 2.02)

= 0.0217

(c)

$\sigma$ _{$\bar x$} = $\sigma$ / $\sqrt$ n = 325 / $\sqrt$ 25 = 65

= P[(1550 - 1518) / 65 < ( $\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (1575 - 1518) / 65)]

= P(0.492 < Z < 0.877)

= P(Z < 0.877) - P(Z < 0.492)

= 0.1211

(d)

$\sigma$ _{$\bar x$} = $\sigma$ / $\sqrt$ n = 325 / $\sqrt$ 16 = 81.25

= P[(1440 - 1518) / 81.25 < ( $\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (1480 - 1518) / 81.25)]

= P(-0.96 < Z < -0.468)

= P(Z < -0.468) - P(Z < -0.96)

= 0.1514

e)

Original distribution is normal that why we use central limit theorem