Question

Beta Company is manufacturing steel wire with an average tensil and strength of 50 kilos. The laboratory tests 16 pieces and finds that the mean weight is 47 kilos. Are the results in accordance with the hypothesis that the population mean is 50 kilos?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=50$

$H_1:\mu\not=50$

​This corresponds to a two-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 $\alpha=0.05, df=n-1=16-1=15$ degrees of freedom, and the critical value for a two-tailed test is $t_c=2.131449.$

The rejection region for this two-tailed test is $R=\{t:|t|>2.131449\}.$  

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{47-50}{15/\sqrt{16}}=-0.8$$

Since it is observed that $|t|=0.8<2.131449=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value for two-tailed $\alpha=0.05, df=15,$ $t=-0.8$ is $p=0.436198,$ and since $p=0.436198>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population $\mu$ is different than 50, at the $\alpha=0.05$ significance level.