Question

Question 4 [25]
The Zambezi car battery manufacturer claims that the average lifespan of batteries produced by his firm is at least 30 months. The rival manufacture the Rehoboth batteries disagree and took random sample of 100 Zambezi car batteries and recorded a mean of 31.7 months and a standard deviation of 8 months. (Show all your works)
Determine the following:
a) The null and alternative hypotheses

b) The test statistic value.

c) The critical statistics value at 99% confidence level

d) The rejection region using critical value approach

e) The p value at 99% confidence level

f) The rejection region using both the p value approach

g) Make conclusion about the population mean using both approaches

Answer 1

T-test for One Population Mean

The provided sample mean is Xbar=31.7

and the sample standard deviation is s = 8

and the sample size is n = 100.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 30 the average lifespan of batteries produced by his firm is 30 months

Ha: μ > 30 the average lifespan of batteries produced by his firm is at least 30 months

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

(2) Test Statistics

The t-statistic is computed as follows:

(3,4) Critical value and Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a right-tailed test is tc​=2.365.

(t values is calculated using t distribution table with 99 degrees of freedom)

The rejection region for this right-tailed test is R=t:t>2.365

(5) P value

Using the P-value approach: The p-value is p = 0.018

(P value is calculated using t distribution

p value = P[t>2.125] with df = 99

(6) Decision about the null hypothesis

Since it is observed that ]2.125≤tc​=2.365, it is then concluded that the null hypothesis is not rejected.

and since p=0.018≥0.01, it is concluded that the null hypothesis is not rejected.

(7) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ(average lifespan of batteries produced by his firm) is greater than 30, at the 0.01 significance level.