Question

An Izod impact test was performed on 20 specimens of PVC pipe. The ASTM standard for this material requires that Izod impact strength must be greater than 1.0 ft-lb/in. The sample average and standard deviation obtained were x = 1.121 and s = 0.328, respectively. (a) Test H0: µ = 1.0 versus H1: µ 1.0 using α = 0.01 and draw conclusions. (b) Use the P-value approach to confirm your inference (c) Construct the 99% lower confidence bound on the mean and use it to test the hypothesis. (d) Suppose that the mean strength is as high as 1.1 ft-lb/in, the engineer would like to detect this difference with probability at least 0.90. Was the sample size n = 20 used in part (a) adequate? Use the sample standard deviation s as an estimate of σ in reaching your decision. Estimate what sample size is needed to detect this difference.

Answer 1

Given That:-

An Izod impact test was performed on 20 specimens of PVC pipe. The ASTM standard for this material requires that Izod impact strength must be greater than 1.0 ft-lb/in. The sample average and standard deviation obtained were x = 1.121 and s = 0.328, respectively.

(a) Test H0: µ = 1.0 versus H1: µ 1.0 using α = 0.01 and draw conclusions.

Here

vs  

Given

n = 20, s = 0.328

Test statistic

= 1.65

d.f = n - 1

= 20 - 1

= 19

At , critical value is 2.54

Here cal t = 1.65 < 2.54

so do not reject H0.

(b) Use the P-value approach to confirm your inference

here p - value = 0.058 > 0.01

so do not reject H0.

(c) Construct the 99% lower confidence bound on the mean and use it to test the hypothesis.

The lower confidence bound is

= 1.121 - 0.1863

= 0.9347

(d) Suppose that the mean strength is as high as 1.1 ft-lb/in, the engineer would like to detect this difference with probability at least 0.90. Was the sample size n = 20 used in part (a) adequate? Use the sample standard deviation s as an estimate of σ in reaching your decision. Estimate what sample size is needed to detect this difference.

Here error (e) = 1.1 - 1

= 0.1 , s = 0.328

At 90%,  

sample size (x) =

  

= 29.11 30

Here the sample size n = 20 is not adequate we need the sample size n = 30.