Question

calibration curve based on$\mathbf{n}\mathbf{=}\mathbf{10}$known points was used to measure the protein in an unknown. The results were protein $\mathbf{=}\mathbf{15}\mathbf{.}{\mathbf{2}}_{\mathbf{2}}\left(\pm 0.4_6\right)\mathbf{\mu g}$, where the standard uncertainty is ${\mathbf{u}}_{\mathbf{x}}\mathbf{=}\mathbf{0}\mathbf{.}{\mathbf{4}}_{\mathbf{6}\mathbf{}}\mathbf{\mu g}$. Find the90%and 99%confidence intervals for protein in the unknown.

Short answer

90% confidence intervals for protein in an unknown is $15.2\pm 0.8\mathrm{\mu g}$

99% confidence intervals for protein in an unknown is $15\pm 1\mathrm{\mu g}$

Step-by-step solution

Confidence Intervals

Confidence interval $\mathbf{=}\overline{\mathbf{x}}\mathbf{\pm }\frac{\mathbf{ts}}{\sqrt{\mathbf{n}}}\mathbf{=}\overline{\mathbf{x}}\mathbf{\pm }{\mathbf{tu}}_{\mathbf{x}}$since the standard uncertainty

$\left(u_x\right)\mathbf{=}\mathbf{s}\mathbf{/}\sqrt{\mathbf{n}}$

Here,

$\overline{\mathbf{x}}$is the mean,

s is the standard deviation,

t is the Student's t,

uis the number of measurements, and

u_xis the standard uncertainty.

90% confidence intervals for protein

The number of measurements of the known point is $\mathrm{n}=10$

The result of the protein is $15.2_2\left(\pm 0.4_6\right)\mathrm{\mu g}$

The standard uncertainty is ${\mathrm{u}}_{\mathrm{x}}=0.4_6\mathrm{\mu g}$

The numbers in the subscript denotes the insignificant figures.

Calculation of Confidence intervals:

The t value corresponding to 90% confidence level for eight degrees of freedom is ${\mathrm{t}}_{90\%}=1.860$

The t value corresponding to 99% confidence level for eight degrees of freedom is ${\mathrm{t}}_{99\%}=3.355$

90% confidence interval is calculated as follows:

Confidence interval $\overline{\mathrm{x}}\pm {\mathrm{tu}}_{\mathrm{x}}$

90% confidence interval role="math" localid="1663318037622" $=15.22\left(\pm 1.860\times 0.46\right)$

$$\begin{gathered} =15.2_2\left(\pm 0.8_6\right)\mathrm{\mu g} \\ =15.2\left(\pm 0.8\right)\mathrm{\mu g} \end{gathered}$$

The 99% confidence intervals for protein

99% confidence interval is calculated as follows:

Confidence interval$=\overline{\mathrm{x}}\pm {\mathrm{tu}}_{\mathrm{x}}$

90% confidence interval $=15.22\left(\pm 3.355\times 0.46\right)$

$$\begin{gathered} =15.2_2\left(\pm 1.5\right)\mathrm{\mu g} \\ =15\left(\pm 1\right)\mathrm{\mu g} \end{gathered}$$