1. A manufacturer of athletic footwear claims that the mean life of his product will exceed 50 hours. A random sample of 36 pairs of shoes leads to the following results in terms of useful life:

a. Does the sample data provide evidence to conclude that the manufacturer's claim is correct (using a=0.05)? 

Solution:

The information given in the above problem can be represented with the following notations as follows:
Sample size = n = 36
Sample mean = = 52.3
Sample standard deviation = s = 9.6
Population mean = = 50
Significance level = 0.05

1.Null hypothesis:

H₀: 50
That is the mean life of the product is less than 50 hours.

Alternate hypothesis:

H₁: 50

That is, the mean life of the product will exceed 50 hours.

2. Criterion for rejection of the null hypothesis:

If the P-value of the test statistic is less than 0.05 we reject the null hypothesis.

3. Test statistic:

Z =

=

= = 1.4375

Hence the test statistic value is 1.4375

4. P-value:

P-value = P [Z > 1.4375]
= 1 – P [Z <1.4375]
= 1 - 0.924712
= 0.075288
Hence the P- value of the test statistic is 0.075288

5. Statistical decision:

Since the P-value of the test statistic is greater than the significance level (that is 0.075288 > 0.05) we conclude that the sample does not provide enough evidence to reject the null hypothesis at 0.05 level. Thus the mean life of the product does not exceed 50 hours.