Answer to Question #337576 in Statistics and Probability for Yen

Question

Many things can contaminate urban storm water, including discarded batteries. When these batteries rupture, they discharge elements that are harmful to the environment. The sample mean zinc mass of 27 Panasonic AAA batteries was examined and found that it was 2.06g, with a sample standard deviation of 0.141g. Does this data provide compelling evidence for concluding that the population mean zinc mass exceeds 2.0g? Use a significance level α=0.05 to test the hypotheses.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\le 2$

$H_a:\mu>2$

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

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=26$ degres of freedom, and the critical value for a right-tailed test is $t_c = 1.705618.$

The rejection region for this right-tailed test is $R = \{t: t > 1.705618\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{2.06-2}{0.141/\sqrt{27}}\approx2.2111$$

Since it is observed that $t = 2.2111 >1.705618= t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, $df=26$ degrees of freedom, $t=2.2111$ is $p = 0.018014,$ and since $p = 0.018014< 0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is greater than 2, at the $\alpha = 0.05$ significance level.