Question

Suppose that 500 batteries made by a certain manufacturer lasted on the average 2420 hours with a standard deviation of 300 hours. Do you have enough evidence to reject the manufactuer's claim that the average life is at least 2500 hours? Test the claim at significance level a = 0.05. (a) State clearly what your null and alternative hypotheses are. (b) Find the critical region of the test. (c) Find the p-value of the test. (d) What is your conclusion?

Answer 1

Here, we have to use one sample t test for the population mean.

Part a

The null and alternative hypotheses are given as below:

Null hypothesis: H0: the average life is at least 2500 hours.

Alternative hypothesis: Ha: the average life is less than 2500 hours.

H0: µ ≥ 2500 versus Ha: µ < 2500

This is a lower tailed test.

From given data, we have

µ = 2500

Xbar = 2420

S = 300

Part b

n = 500

df = n – 1 = 499

α = 0.05

Critical value = -1.6479

(by using t-table or excel)

Critical region: Reject H0 when t < -1.6479

The test statistic formula is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

t = (2420 - 2500)/[300/sqrt(500)]

t = -5.9628

Part c

P-value = 0.0000

(by using t-table)

Part d

P-value < α = 0.05

So, we reject the null hypothesis

There is not sufficient evidence to conclude that the average life is at least 2500 hours.