Question

Suppose x has a distribution with a mean of 70 and a standard deviation of 52. Random samples of size n = 64 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the distribution. x has distribution with mean μx = and standard deviation σx = .

(b) Find the z value corresponding to x = 83. z =

(c) Find P(x < 83). (Round your answer to four decimal places.) P(x < 83) =

(d) Would it be unusual for a random sample of size 64 from the x distribution to have a sample mean less than 83? Explain.

a. Yes, it would be unusual because less than 5% of all such samples have means less than 83.

b. No, it would not be unusual because more than 5% of all such samples have means less than 83.

c.Yes, it would be unusual because more than 5% of all such samples have means less than 83.

d. No, it would not be unusual because less than 5% of all such samples have means less than 83.

Answer 1

Solution :

Given that ,

mean = = 70

standard deviation = = 52

n = 64

The sampling distribution of mean and standard deviation is ,

= 70 and

= / n = 52 / 64 = 52 / 8 = 6.5

(b)

x = 83

z = (83 - 70) / 52 = 13 / 52 = 0.25

P(z < 0.25) = 0.5987

Probability = 0.5987

P( < 83) = P(( - ) / < (83 - 70) / 6.5)

= P(z < 2)

= 0.0228

  

a. Yes, it would be unusual because less than 5% of all such samples have means less than 83.