Question

A random sample of size n₁ = 14 is selected from a normal population with a mean of 74 and a standard deviation of 6. A second random sample of size n₂ = 9 is taken from another normal population with mean 70 and standard deviation 14. Let X¯1and X¯2 be the two sample means. Find:

(a) The probability that X¯1-X¯2 exceeds 3.

(b) The probability that 4.4 ≤X¯1-X¯2≤ 5.4.

Answer 1

Let , and n1=14

Also , and n2=9

Now , the sampling distribution of sample mean is ,

Therefore , the probability distribution of is also a normal distribution with mean and standard deviation

Where ,

(a) Now ,

; From the standard normal probability table

The probability that   exceeds 3 is 0.5636

(b)

Now ,

; From the standard normal probability table

The probability that 4.4 ≤≤ 5.4 is 0.0632.