A shoemaker company produces a specific model of shoes having 15 months average lifetime. One of the employee in their R & D division claims to have developed a product that lasts longer. This latest product was worn by 30 people and lasted on average for 17 months. The variability of the original shoes is estimated based on the standard deviation of new group which is 5.5 months. Is the designer's claim of a better shoe supported by the findings of the trial? Make your decision using two tailed testing using a level of significance 0.05.

No, the claim is not supported; there is no significant difference in lifetime.

The correct answer is Option (2) → No, the claim is not supported; there is no significant difference in lifetime.

Given $μ_0= 15$ months, $\bar x = 17$ months, $n = 30$ and $S = 5.5$

Let the hypothesis be given as

Null hypothesis $H_0: μ = 15$

Alternative hypothesis $H_a:  μ ≠ 15$

So, the test statistic $t =\frac{\bar x - μ_0}{\frac{S}{\sqrt{n}}}=\frac{17-15}{\frac{5.5}{\sqrt{30}}}=\frac{2×\sqrt{30}}{5.5}$

$⇒t = 1.9917$

$df=30-1=29$

$∵t = 1.9917 > 0$.

$t_{α/2}=t_{0.025}$ at $df = 29$ is $t_{0.025} = 2.045$

$∵1.9917 < 2.045 (t <t_{α/2})$

so, do not reject $H_0$.

Hence, No, the claim is not supported; there is no significant difference in lifetime.