Question

6. In 2005, the distribution of the score in the math portion of the SAT test was approximately normal with a mean y = 520 and a standard deviation o = 115. (a) What percentage of students scored more than 720 in the math portion of the SAT test in 2005? (b) Suppose that sample of 64 students that took the SAT in 2005 is randomly selected. What is the probability that the average score x in the math portion of the SAT for these 64 students is between 491 and 549?

Answer 1

Solution :

mean = $\mu$ = 520

standard deviation = $\sigma$ =115

a) P(x > 720 ) = 1 - p( x< 720 )

=1- p [(x - $\mu$ ) / $\sigma$ < (720 - 520) /115 ]

=1- P(z < 1.74)

= 1 - 0.9591 = 0.0409

probability = 0.0409

b)

n = 64

$\mu$$\bar x$ =$\mu$ = 520

$\sigma$$\bar x$ = $\sigma$ / $\sqrt$ n = 115/ $\sqrt$ 64 = 14.375

P(491<  $\bar x$ < 549)  

= P[(491 - 520) /14.375< ( $\bar x$ - $\mu$ $\bar x$) / $\sigma$ $\bar x$ < (549 -520) /14.375 )]

= P( -2.02 < Z < 2.02 )

= P(Z <2.02) - P(Z < -2.02 )

= 0.9783 - 0.0217 = 0.9566

probability = 0.9566