A manufacturer of bicycle tires has developed a new design which he claims has an average lifespan of 15 years with standard deviation of 2 years. A dealer of the product claims that the average lifespan of 150 samples of the tires is only 4 years. Test the significant difference of the population and sample means.

$n=150\\\bar x=4\\\sigma=2$

Hypotheses

$H_0:\mu=15\\vs\\H_1:\mu\not=15$

The test statistic is,

$Z={\bar x-\mu\over{\sigma\over\sqrt{n}}}={4-15\over {2\over\sqrt{150}}}=-67.36$

The critical value is,

$Z_{0.05\over2}=Z_{0.025}=-1.96$

Reject the null hypothesis if, $Z\lt Z_{0.025}$

Since $Z=-67.36\lt Z_{0.025}=-1.96$, the null hypothesis is rejected and conclude that the data provide sufficient evidence to show that there is a significant difference between the population mean and the sample mean at 5% level of significance.