Question

A random sample of n = 100 observations is drawn from a population with mean 10 and variance 400.

(a) Describe the shape of the sampling distribution of ¯x. Does your answer depend on the sample size? (

b) Give the mean and standard deviation of the sampling distribution of ¯x.

(c) Find the probability that ¯x is less than 8.

(d) Find the probability that ¯x is greater than 9.5.

(e) Find the probability that ¯x is between 8 and 9.5.

Answer 1

Solution :

(a)

The shape of the sampling distribution of is approximately normal .

Yes

(b)

The sampling distribution of mean is ,

= 10

The sampling distribution of standard deviation is ,  

= / n = 20 / 100 = 2

(c)

P( < 8) = P(( - ) / < (8 - 10) / 2)

= P(z < -1)

= 0.0228

(d)

P( > 9.5) = 1 - P( < 9.5)

= 1 - P[( - ) / < (9.5 - 10) / 2]

= 1 - P(z < -0.25)

= 0.5987

(e)

P(8 < x bar < 9.5) = 0.4013 - 0.0228 = 0.3785