Question

5.2.17 Use the normal distribution of SAT critical reading scores for which the mean is 511 and the standard deviation is 117 . Assume the variable x is normally distributed. left parenthesis a right parenthesis What percent of the SAT verbal scores are less than 650 ​? left parenthesis b right parenthesis If 1000 SAT verbal scores are randomly​ selected, about how many would you expect to be greater than 575 ​? left parenthesis a right parenthesis Approximately nothing ​% of the SAT verbal scores are less than 650 . ​(Round to two decimal places as​ needed.)

Answer 1

We know that the SAT scores are normally distributed.

Mean= 511

Standard deviation= 117

a.

P(score<650)= P(Z<(650-511)/117)

=P(Z<1.188034)

=P(Z<1.19) (This value can be found from a z distribution table)

Thus, percentage of values below 650= 0.8830

b.

We know that a sample size of 1000 is selected.

According to the CLT, the mean of the new distribution will be the same of the parent distribution, and the standard deviation will be equal to the old standard deviation, divided by the square root of the number of samples.

Thus,

P(mean score>575)

= 1-P(mean score<575)

= 1- P(Z<(575-511)/117/sqrt(1000))

= 1-P(Z<64/3.699865)

= 1-P(Z<17.29793) (This is a very high value, almost close to 1)

= 1- 1

= 0