Question

# In replicate analyses, the carbohydrate content of a glycoprotein ( a protein with sugars attached to it) is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohydrate per 100 g of protein. Find the 90% confidence intervals for the carbohydrate content.

Confidence intervals
In replicate analyses, the carbohydrate content of a glycoprotein ( a protein with sugars attached to it) is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohydrate per 100 g of protein. Find the 90% confidence intervals for the carbohydrate content.

2020-12-04

Step 1 Data values are given population standard deviation is not known hence we use T Interval. Sample size $$\displaystyle{\left({n}\right)}={5}$$ Degrees of freedom(df): $$\displaystyle{d}{f}={n}-{1}$$
$$\displaystyle={5}-{1}$$
$$\displaystyle={4}$$ Confidence level $$=0.90$$ Use T table or excel command =T.INV.2T $$\displaystyle{\left({1}-{0.90},{4}\right)}$$ to get critical value Critical value $$\displaystyle{\left({t}_{{{c}}}\right)}={2.132}$$

$$\begin{array}{|c|c|} \hline & X & (X−12.54)^{2} \\ \hline & 12.6 & 0.0036 \\ \hline & 11.9 & 0.4096 \\ \hline & 13 & 0.2116 \\ \hline & 12.7 & 0.0256 \\ \hline & 12.5 & 0.0016 \\ \hline Total & 62.7 & 0.652 \\ \hline \end{array}$$

Sample mean: $$\displaystyle\overline{{{x}}}={\frac{{{\sum_{{{i}={1}}}^{{5}}}\xi}}{{{n}}}}$$
$$\displaystyle={\frac{{{62.7}}}{{{5}}}}$$
$$\displaystyle{12.54}$$ Sample standard deviation: $$\displaystyle{s}=\sqrt{{{\frac{{{\sum_{{{i}={1}}}^{{5}}}{\left({x}_{{{i}}}-\overline{{{x}}}\right)}^{{{2}}}}}{{{n}-{1}}}}}}$$
$$\displaystyle=\sqrt{{{\frac{{{\sum_{{{i}={1}}}^{{5}}}{\left({x}_{{{i}}}-{12.54}\right)}^{{{2}}}}}{{{5}-{1}}}}}}$$
$$\displaystyle=\sqrt{{{\frac{{{0.652}}}{{{4}}}}}}$$
$$\displaystyle=\sqrt{{{0.163}}}$$
$$\displaystyle={0.403733}$$ Step 3 $$\displaystyle=\overline{{{x}}}\pm{t}_{{{c}}}{\frac{{{s}}}{{\sqrt{{{n}}}}}}$$
$$\displaystyle={12.54}\pm{2.132}\times{\frac{{{0.403733}}}{{\sqrt{{{5}}}}}}$$
$$\displaystyle={12.54}\pm{2.132}\times{0.180555}$$
$$\displaystyle={12.54}\pm{0.3849}$$
$$\displaystyle={\left({12.54}-{0.3849},{12.54}+{0.3849}\right)}$$
$$\displaystyle={\left({12.1551},{12.9249}\right)}$$

Step 4 Answer: $$90\%$$ confidence interval is ( 12.16, 12.92 )