Question

The diameters of aluminum alloy rods produced on an extrusion machine are known to have a standard deviation of 0.0001 in. A random sample of 25 rods has an average diameter of 0.5046 in.

(a) Test the hypothesis that mean rod diameter is 0.5025 in. Assume a two-sided alternative and use

(b) Find the P-value for this test.

(c) Construct a 95% two-sided confidence interval on the mean rod diameter.

Answer 1

Solution:

Given ,

= 0.0001

n = 25

= 0.5046

Claim: mean rod diameter is 0.5025 in.

a) Using the claim , the hypothesis can be written as

H0: μ = 0.5025  vs   H1: μ ≠ 0.5025

≠ sign indicates two sided alternative

Since the population SD() is known , we use z test.

The test statistic z is given by

z =

= ( 0.5046 - 0.5025)/( 0.0001/25)

= 105

Let , α=0.05

For two sided hypothesis , the critical value are

  ^{= z}0.025  = 1.96 (use z table)

Modulus of 105 is greater than modulus of 1.96

So, we reject H0 and conclude that  mean rod diameter is significantly different from 0.5025 in.

b) Find p value.

For two tailed test ,

p value = 2 * P(Z < -z) = 2 * P(Z < -105) = 2 * 0 = 0

p value is 0.0000

c) 95% confidence interval for the mean

Here c = 95% = 0.95

= 1 - c = 0.05

  ^{= z}0.025  = 1.96 (use z table)

Confidence interval is given by

     *  

   0.5046   1.96 * 0.0001/25

0.5046     0.0000392

(0.5045608 , 0.5046392)