HypothesisHypothesisTestingTestingforforMeansMeans--SmallSmallSamplesSamples
Do US bathroom scales underestimate a person`s true weight? A $75 \mathrm{~kg}$ test weight was placed on end of $8$ randomdly selected bathroom scales. The readings were as follows:$$\begin{array}{8*{c}} 76.1 & 74.8 & 74.9 & 73.6 & 75.4 & 72.8 & 74.1 & 74.0 \end{array} $$
For the population of bathroom scales test $H_0: \mu=75$ versus $H_1: \mu<75$ at $\alpha=0.05$ level of significance. Report the $p$-value and draw a conclusion in the context of the problem.
$ \begin{aligned} \bar{x}&=74.475 \quad s=1.063 \end{aligned} $$$ t=\frac{74.475-75}{1.063/\sqrt{8}}=-1.40 \quad \quad (df=n-1=7)$$$0.10 \leq p-\text{value} \leq 0.125$.
Fail to reject $H_0$. This sample does not provide enough evidence to claim that the bathroom scales undersestimate weight.
Dichluro-Diphenyl-Toichloroethane (DDT) is still used in some regions of the world as an agricultural insecticide. DDT causes cancer, reproductive problems, and liver damage.
A study of $6$ ponds in south-east Asia found the following readings for the concentration of DDT (in ppm)$$ \begin{array}{6*c} 9.5 & 9.6 & 9.3 & 9.5 & 9.7 & 9.2\end{array}$$At $\alpha=0.05$ level of significance test $H_0: \mu=9.0 $ versus $H_1: \mu>9.0 \mathrm{~ppm} $
$ \begin{aligned} \bar{x}&=9.467 \quad s=0.1862 \end{aligned}$$$t=\frac{9.476-9}{0.1862/\sqrt{6}}=6.14 \quad \quad (df=n-1=5)$$$0.0005 \leq P-value \leq 0.005$.
Reject (strongly) $H_0$. The concentration of DDT in this region exceeds $9 \mathrm{~ppm}$.