Question

In a recent year, the scores for the reading portion of a test were normally distributed, with a mean of 20.2 and a standard deviation of 5.4. 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 . (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 17.2 and 23.2. The probability of a student scoring between 17.2 and 23.2 is (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 31.2. The probability of a student scoring more than 31.2 is (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.

Answer 1

Part a)

X ~ N ( µ = 20.2 , σ = 5.4 )
P ( X < 16 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 16 - 20.2 ) / 5.4
Z = -0.7778
P ( ( X - µ ) / σ ) < ( 16 - 20.2 ) / 5.4 )
P ( X < 16 ) = P ( Z < -0.7778 )
P ( X < 16 ) = 0.2183

Part b)

X ~ N ( µ = 20.2 , σ = 5.4 )
P ( 17.2 < X < 23.2 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 17.2 - 20.2 ) / 5.4
Z = -0.5556
Z = ( 23.2 - 20.2 ) / 5.4
Z = 0.5556
P ( -0.56 < Z < 0.56 )
P ( 17.2 < X < 23.2 ) = P ( Z < 0.56 ) - P ( Z < -0.56 )
P ( 17.2 < X < 23.2 ) = 0.7108 - 0.2892
P ( 17.2 < X < 23.2 ) = 0.4215

Part c)

X ~ N ( µ = 20.2 , σ = 5.4 )
P ( X > 31.2 ) = 1 - P ( X < 31.2 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 31.2 - 20.2 ) / 5.4
Z = 2.037
P ( ( X - µ ) / σ ) > ( 31.2 - 20.2 ) / 5.4 )
P ( Z > 2.037 )
P ( X > 31.2 ) = 1 - P ( Z < 2.037 )
P ( X > 31.2 ) = 1 - 0.9792
P ( X > 31.2 ) = 0.0208

part d)

If the probability is less than 0.05, the it is called unusual event

Probability of part C is less than 0.05, hence it is unusual.