Question

Assume that in a given year the mean mathematics SAT score was 499, and the standard deviation was 112. A sample of 63 scores is chosen.

a) What is the probability that the sample mean score is less than 481? Round the answer to at least four decimal places.

b) What is the probability that the sample mean score is between 462 and 504? Round the answer to at least four decimal places.

c) Find the 55^th percentile of the sample mean. Round the answer to at least two decimal places.

d) Would it be unusual if the sample mean were greater than 524? Round answer to at least four decimal places. It would/would not be unusual if the sample mean were greater than 524 since the probability is...?

e) Do you think it would be unusual for an individual to get a score greater than 524? Explain. Assume the variable is normally distributed. Round the answer to at least four decimal places. Yes/No, because the probability that an individual gets a score greater than 524 is...

Answer 1

Solution :

mean = = 499

standard deviation = = 112

n = 63

= = 499

= / n = 112/ 63 = 14.1107

a) P( < 481) = P(( - ) / < (481 -499) / 14.1107)

= P(z < -1.28 )

= 0.1003

probability = 0.1003

b)

P( 462< x <504 ) = P[(462 -499 )/14.1107 ) < (x - ) /  < (504 -499) /14.1107 ) ]

= P(-2.62 < z < 0.35)

= P(z <0.35 ) - P(z <-2.62 )

Using standard normal table

= 0.6368 - 0.0044 = 0.6324

Probability = 0.6324

c)

P(Z < z) = 0.55

z = 0.126

Using z-score formula,

= z * +   

=0.126 * 14.1107+ 499 = 500.78

d)

P( >524 ) = 1 - P( < 524 )

= 1 - P[( - ) / < (524 -499) /14.1107 ]

= 1 - P(z < 1.77 )

= 1 - 0.9616 = 0.0384

Probability = 0.0384

It would be unusual .

The sample mean were greater than 524 since the probability is = 0.0384

e)

P(x >524 ) = 1 - p( x< 524)

=1- p [(x - ) / < (524 -499) /112]

=1- P(z <0.22 )

= 1 - 0.5871 = 0.4129

probability = 0.4129

No .  because the probability that an individual gets a score greater than 524 is greater than 0.05.