Question

A manufacturer of running shoes knows that the average lifetime for a particular model of shoes is 15 months. Someone in the research and development division of the shoe company claims to have developed a longer lasting product. This new product was worn by 25 individuals and lasted on average for 17 months. The variability of the original shoe is estimated based on the standard deviation of the new group which is 5.5 months. Is the designer's claim of a better shoe supported by the trial results?

Answer 1

The decision rule is

Fail to reject the null hypothesis

The conclusion is

There is no sufficient evidence to show that the designer's claim of a better shoe is supported by the trial results.

Explanation:

From the question we are told that

The population mean is
`\mu = 15 \ months`

The sample size is n = 25

The sample mean is
`\= x = 17 \ months`

The standard deviation is
`s = 5.5 \ months`

Let assume the level of significance of this test is
`\alpha = 0.05`

The null hypothesis is
`H_o : \mu = 15`

The alternative hypothesis is
`H_a : \mu > 17`

Generally the degree of freedom is mathematically represented as

`df = n -1`

=>
`df = 25 -1`

=>
`df = 24`

Generally the test statistics is mathematically represented as

`t = (\= x- \mu )/((s)/(√(n) ) )`

=>
`t = ( 17 - 15 )/((5.5)/(√(25) ) )`

=>
`t = 1.8182`

Generally from the student t distribution table the probability of obtaining
`t = 1.8182` to the right of the curve at a degree of freedom of
`df = 24` is

`p-value = P(t > 0.18182 ) = 0.4286`

From the value obtained we see that
`p-value > \alpha` hence

The decision rule is

Fail to reject the null hypothesis

The conclusion is

There is no sufficient evidence to show that the designer's claim of a better shoe is supported by the trial results.