Question

In a recent​ year, the scores for the reading portion of a test were normally​ distributed, with a mean of

22.7

and a standard deviation of

6.3.

Complete parts​ (a) through​ (d) below.

​(a) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is less than

16.

The probability of a student scoring less than

16

is

nothing.

​(Round to four decimal places as​ needed.)

​(b) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is between

13.9

and

31.5.

The probability of a student scoring between

13.9

and

31.5

is

nothing.

​(Round to four decimal places as​ needed.)

​(c) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is more than

35.7.

The probability of a student scoring more than

35.7

is

nothing.

​(Round to four decimal places as​ needed.)

​(d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.

Answer 1

Solution:- Given that mean = 22.7, sd = 6.3

(a) P(X < 16) = P((X-mean)/sd < (16-22.7)/6.3 )
= P(Z < -1.0635)
= 1 - P(Z < 1.0635)
= 1 - 0.8554
= 0.1446

(b) P(13.9 < X < 31.5) = P((13.9-22.7)/6.3 < Z < (31.5-22.7)/6.3 )
= P(-1.3968 < Z < 1.3968)
= P(Z < 1.3968) - P(Z < -1.3968)
= 0.9192 - 0.0808
= 0.8384

(c) P(X > 35.7) = P(Z > (35.7-22.7)/6.3)
= P(Z > 2.0635)
= 0.0197

  
(d) event in part (c) is unusual .