The problem proves that the distributor's claim is true and he has a right to complain.
But when the confidence interval is changed from 95% to 90% the distributor's claim proves to be false.
The Confidence Interval for 95% is 0.9687;0.98532
For a sample size of n>30 the central limit theorem allows us to assume that the sampling distribution of x~ is approximately normal.
The critical region is Z ≥ 1.28 therefore the 1.877 lies in the rejection region that the distributor's claim is not true.
The Confidence Interval for 90% is 0.97156;0.982431
Here
μ = 0.977
s= σ= 0.03
95 % confidence interval is given by
x~± z∝/2 (s/√n)
Putting the values
0.977 ± 1.96 (0.03/√50)
=0.977 ± 0.008315
0.9687;0.98532
Part B:
x~+ 0.95= 0.977+0.95= 1.927
The critical region is Z ≥ 1.96 therefore the 1.927 lies in the acceptance region that the distributor's claim is true.
Part C:
For a sample size of n>30 the central limit theorem allows us to assume that the sampling distribution of x~ is approximately normal.
Part D:
μ = 0.977
s= σ= 0.03
90 % confidence interval is given by
x~± z∝/2 (s/√n)
Putting the values
0.977 ± 1.28 (0.03/√50)
=0.977 ± 0.005431
0.97156;0.982431
Part D:
x~+ 0.90= 0.977+0.90= 1.877
The critical region is Z ≥ 1.28 therefore the 1.877 lies in the rejection region that the distributor's claim is not true.
For further understanding of acceptance and rejection region click
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