statistical testing

Question

The manufacturer of a particular brand of tires claims they average at least 50,000 miles before needing to be replaced. From past studies of this tire, it is known that the population standard deviation is 8,000 miles.

A survey of tire owners was conducted. From the 33 tires surveyed, the mean lifespan was 45000 miles. Using alpha = 0.05, can we prove that the data in inconsistent with the manufacturers claim?

Test Statistic=?

p-value=?

Answer 1

Null Hypothesis: mean = 50000

Alternate Hypothesis: mean != 50000

this is a two sided test for a single mean. Since the population standard deviation is know we can use the z-statistic:

z = (sample mean - population mean)/(population standard deviation/square root(sample size)) =

(45000-50000)/(8000/sqrt(33)) = -3.59

since a two-sided test

p-value = P(z > 3.59) + P(z < -3.59)

since normal distribution is symmetric: p-value = 2*P(z < -3.59) = 2 * 0.0002 = 0.0004.

p-value is the lowest value of alpha at which you can reject the null hypothesis. since p-value is less than 0.0004 we can reject the null hypothesis.