Hypothesis: Sample Size Alpha

Case No 7
Tiresome Tire I

Introduction: This case is about Hypothesis Testing, there is a tire manufacturing company which is producing tire with the strength 2800 pound per square inch (psi), we have to test Null Hypothesis i.e. H0 : u> 2,800 psi, where u is the mean strength of large batch of tire and population SD is 10 psi.

Calculation:
We have to test Null Hypothesis based on given information.
Type I error will result in the rejection of a large number of good tires.
Type II error of passing a faulty batch of tires can result in fatal accidents on the roads.

Alpha = 5%, Sample Size = 40, for calculating Beta u = 2790 psi.

H0 : u> 2,800
Test Hypothesis
Sigma
10
Sample Size
40
Alpha
0.05
Z alpha
-1.644853627
Z calculated
2797.399258
X bar
2790
Z critical
4.679701693
Power
0.999998564
Beta
0.000001436

Calculate Power and Beta for the sample size 30, 40, 60 and 80. Alpha = 5%.

Beta(β) at different sample size with alpha 0.5
There are two methods for calculating Beta, 1st method: Direct use NORMINV(α, u, (σ/√n)) for Z calculation and NORMDIST(Zcal, X', (σ/√n),TRUE) for calculating Power.

Sample Size
Alpha
X bar
Sigma
Z
Power
Beta

30
0.05
2790
10
2796.996922
0.999936543
0.000063457
40
0.05
2790
10
2797.399258
0.999998564
0.000001436
60
0.05
2790
10
2797.876503
0.999999999
0.000000001
80
0.05
2790
10
2798.160998
1.000000000
0.000000000
2nd Method: first we find Zcal with formula [u ± Zα*(σ/√n)], after that find Zcritical with formula [(Zcal - X')/ (σ/√n)], than calculate Power using NORMSDIST(Zcritical).

Sample Size
Alpha
X bar
Z cal
Z critical
Power
Beta
30
0.05
2790
2796.99692
3.832371948
0.999936543
0.000063457
40
0.05
2790
2797.39926
4.679701693
0.999998564
0.000001436
60
0.05
2790
2797.8765
6.101113065
0.999999999
0.000000001
80
0.05
2790
2798.161
7.299418283
1.000000000
0.000000000

Question 1: Calculate the exact