A variable X follows a normal distribution with mean 20 and standard deviation 3. Approximately 95% of the distribution can be found between what values of X?
Select one:
a. 0 and 23
b. 17 and 23
c. 18 and 22
d. 14 and 26
e. 10 and 30
Solution:-
Given that,
mean = = 20
standard deviation = = 3
Using standard normal table,
P( -z < Z < z) = 95%
= P(Z < z) - P(Z <-z ) = 0.95
= 2P(Z < z) - 1 = 0.95
= 2P(Z < z) = 1 + 0.95
= P(Z < z) = 1.95 / 2
= P(Z < z) = 0.975
= P(Z < 1.96 ) = 0.975
= z ± 1.96
Using z-score formula,
x = z * +
x = -1.96 * 3 + 20
x = 14.12
x = 14
Using z-score formula,
x = z * +
x = 1.96 * 3 + 20
x = 25.88
x = 26
The middle 95% are from 14 and 26
correct option is = d
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