Question

A student in stat 3113 class is investigating lifetime of a new bulb he invented. He has built 10 bulbs and tested them to end-of-life in a test. The sample mean and sample standard deviation are 950 hrs and 230 hrs, respectively. The student would like to test whether that the mean life of this new bulb is greater than industry average 700 hrs. Use $\alpha$ =0.05

a) What type of test would be suitable to test this hypotheses?

b) Write the proper hypothesis

c) Test the hypothesis using Critical value approach

d) Test the hypothesis using p value approach

Answer 1

a) What type of test would be suitable to test this hypothesis?.

Since population variance is unknown and the sample size is less than 30. we will use onr sample t-test.

The provided sample mean is$\bar X$ = 950 and the sample standard deviation is s = 230, and the sample size is n = 10.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: $\mu$ ≥ 700

The lifetime of bulb is more than the industry average, 700.

Ha: $\mu$ < 700

The lifetime of bulb is less than the industry average, 700.

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation, will be used.

(2) Rejection Region

Based on the information provided, the significance level is
= 0,05
, and the critical value for a left-tailed test is
te = -1.833
.

(3) Test Statistics

The t-statistic is computed as follows:

X-HO s/n -

ta 950 - 700 230/10

t = 3.437

(4) Decision about the null hypothesis

Since it is observed that t = 3.437 >$t_c$ = -1.833, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.9963, and since p = 0.9963 > 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean $\mu$ is less than 700, at the 0.05 significance level.

Hence, the bulb he made has the lifetime more than the industry average, which is 700.