Question

A sample of size 42 will be drawn from a population with mean 52 and standard deviation 11.

(a) Is it appropriate to use the normal distribution to find probabilities for x(bar)?

(b) If appropriate find the probability that x(bar) will be between 53 and 54. Round the answer to at least four decimal places.

(c) If appropriate find the 46th percentile of x(bar). Round the answer to at least two decimal places.

Answer 1

Solution :

Given that,

mean = = 52

standard deviation = = 11

n = 42

a) Yes. The probability distribution of x is approximately normal with

= = 52

= / n = 11 / 42 = 1.6973

b) P(53 < < 54 )  

= P[(53 -52) /1.6973 < ( - ) / < (54 -52) /1.6973 )]

= P( 0.59< Z < 1.18 )

= P(Z < 1.18) - P(Z < 0.59)

Using standard normal table

= 0.881 - 0.7224 = 0.1586

probability = 0.1586

c) 46% = 0.46

P(Z < z) = 0.46

z = -0.10

= z * + = (-0.10)*1.6973+52

= 51.83