Question 14.8: The two samples analyzed in Example 14.7 are known to contai...
The two samples analyzed in Example 14.7 are known to contain the following concentrations of cholesterol.
\mu_{\text {samp 1 }}=248.3~ \mathrm{mg} / 100 \mathrm{~mL} \quad \mu_{\text {samp 2 }}=247.6~ \mathrm{mg} / 100 \mathrm{~mL}
Determine if there is any evidence for a systematic error in the method at the 95% confidence level.
Using the data from Example 14.7 and the true values for the samples, we know that s_{\mathrm{T}} is 13.3 , and
\begin{gathered} \bar{T}=\bar{X}_{\text {samp } 1}+\bar{X}_{\text {samp 2 }}=245.9+243.5=489.4 \\ \mu_{\text {tot }}=\mu_{\text {samp } 1}+\mu_{\text {samp 2 }}=248.3+247.6=495.9 \end{gathered}
Substituting these values into equation 14.20
t_{\mathrm{exp}}\,=\,{\frac{\left|\overline{{{T}}}\,-\,\mu_{\mathrm{tot}}\right|\sqrt{n}}{s_{\mathrm{T}}\sqrt{2}}}\,. (14.20)
gives
t_{\exp }=\frac{|489.4-495.9| \sqrt{10}}{13.3 \sqrt{2}}=1.09
This value for t_{\exp } is smaller than the critical value of 2.26 for t(0.05,9). Thus, there is no evidence for a systematic error in the method at the 95 \% confidence level.