Question

in a study of distances traveled by buses before the first major engine filure a sample of 191 buses resulted in mean of 96700 miles and a stndard deviation of 37500 mules.at the 0.05 level of significance test the manufcturer's claim that the mean distance traveled before major engine failure is more than 90000 miles.

Answer 1

We have that:

$n = 191$

$\bar x = 96700$

$s = 37500$

$\alpha = 0.05$

$\mu=90000$

$H_0:\mu = 90000$

$H_a:\mu>90000$

The hypothesis test is right-tailed.

Since the population standard deviation is unknown we use the t-test.

The critical value at the 5% significance level is and 190 df is 1.65

(degrees of freedom df = n – 1 = 191 – 1 = 190)

The critical region is t > 1.65

Test statistic:

$$t=\frac{\bar x-\mu}{\frac{s}{\sqrt n}}=\frac{96700-90000}{\frac{37500}{\sqrt 191}}=2.47$$

Since 2.47 > 1.65 thus t falls into rejection region therefore we reject the null hypothesis.

At the 5% significance level the data do provide sufficient evidence to confirm the manufacturer's claim. We are 95% confident to conclude that the mean distance traveled before a major engine failure is more than 90,000 miles.