Question

given a normal distribution with µ = 105 and Q=25 , if you select a sample of n=25 ,what is the probability that X is
a, less than 94?
b, between 95 and 95.5?
c, above 106.2?
d, there is a 60% chance that X is above what value?

σ=25

Answer 1

Standard error of mean = = 25 / = 5

By Central limt theorem,

a,

P( < 94) = P[Z < (94 - 105)/5] = P[Z < -2.2] = 0.0139 (Using Z distribution table)

b.

P(95 < < 95.5) = P( < 95.5) - P( < 95)

= P[Z < (95.5 - 105)/5] - P[Z < (95 - 105)/5]

= P[Z < -1.9] - P[Z < -2]

= 0.0287 - 0.0228 (Using Z distribution table)

= 0.0059

c.

P( > 106.2) = P[Z > (106.2 - 105)/5] = P[Z > 0.24] = 0.4052 (Using Z distribution table)

d.

Let k be the value such that P(X > k) = 0.60

=> P(X < k) = 1 - 0.60 = 0.40

Z score for 40th percentile is -0.2533

k = 105 - 0.2533 * 5 = 103.7335

Thus, X is above 103.7335 with 60% chance.