Answer to Question #243523 in Statistics and Probability for smilynne

Question

Powder milk is packed in 1-kilogram bag. An inspector from the Department of Trade and Industry (DTI) suspects that bags may not contain 1 kilogram. A sample of 40 bags produces a mean of 0.96 kilograms and a standard deviation of 0.12 kilogram. Is there enough evidence to conclude that the bags do not contain 1 kilogram as stated at 0.05 level of confidence?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\geq1$

$H_1:\mu<1$

This corresponds to a left-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 $\alpha = 0.05,$

$df=n-1=40-1=39$ degrees of freedom, and and the critical value for a left-tailed test is $t_c= -1.684875.$

The rejection region for this left-tailed test is $R = \{t: t < -1.684875\}.$

The t-statistic is computed as follows:

$$t-=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{0.96-1}{0.12/\sqrt{40}}=-2.108185$$

Since it is observed that $t = -2.108185 < -1.684875=t_c ,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for left-tailed $\alpha=0.05, df=39,$ $t=-2.108185$ is $p=0.020746,$ and since $p=0.020746<0.05=\alpha,$

it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$  is less than $1,$ at the $\alpha = 0.05$ significance level.

Therefore, there is enough evidence to claim that the bags do not contain 1 kilogram as stated, at the $\alpha = 0.05$ significance level.