Question

A poll used a sample of 100 randomly selected car owners. Within the​ sample, the mean time of ownership for a single car was 5.89 years with a standard deviation of 3.26 years. Test the claim by the owner of a large dealership that the mean time of ownership for all cars is less than 6.5 years. Use a 0.05 significance level.

Answer 1

We reject the null hypothesis at the significance level of 0.05.

Explanation:

`H_(0): \mu = 6.5` vs
`H_(1): \mu < 6.5` (lower-tail alternative)

We have
`\bar{x} = 5.89`,
`\sigma = 3.26` and n = 100. We have a large sample and our test statistic is

`Z = \frac{\bar{X}-6.5}{\sigma/√(n)}` which is normal standard approximately. We have observed

`z = (5.89-6.5)/(3.26/√(100)) = -1.8712`.

We should use the significance level
`\alpha = 0.05`. The 5th quantile of the standard normal distribution is
`z_(0.05) = -1.6449` and the rejection region is given by RR = z < -1.6449. Because the observed value -1.8712 is less than -1.6449, we reject the null hypothesis at the significance level of 0.05.