Question

A population of values has a normal distribution with μ = 194 μ = 194 and σ = 13.4 σ = 13.4 . You intend to draw a random sample of size n = 127 n = 127 . Find the probability that a single randomly selected value is greater than 197.3. P(X > 197.3) = Find the probability that a sample of size n = 127 n = 127 is randomly selected with a mean greater than 197.3. P(M > 197.3) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Solution :

Given that ,

mean = = 194

standard deviation = = 13.4

a) P(x > 197.3) = 1 - p( x< 197.3)

=1- p P[(x - ) / < (197.3 - 194) / 13.4]

=1- P(z < 0.246)

Using z table,

= 1 - 0.5972

= 0.4028

b) n = 127

=    = 194

= / n = 13.4 / 127 = 1.189

P(M > 197.3) = 1 - P(M < 197.3)

= 1 - P[(M - ) / < (197.3 - 194) / 1.189]

= 1 - P(z < 2.775)

Using z table,    

= 1 - 0.9972

= 0.0028