Question

Rainwater was collected in water containers at $30$different sites near an industrial complex, and the amount of acidity (pH level) was measured. The mean and standard deviation of the values are $4.60$and $1.10,$respectively. When the $pH$ meter was recalibrated back at the laboratory, it was found to be in error. The error can be corrected by adding $0.1$ pH unit to all of the values and then multiplying the result by $1.2$What are the mean and standard deviation of the corrected $pH$ measurements?

$$\begin{gathered} a.5.64,1.44 \\ b.5.64,1.32 \\ c.5.40,1.44 \\ d.5.40,1.32 \\ e.5.64,1.20 \end{gathered}$$

Short answer

The correct option is (b)$5.64,1.32$

Step-by-step solution

Given information

$$\begin{gathered} \mu x=4.60 \\ \sigma x=1.10 \end{gathered}$$

Concept

Properties of mean and standard deviation

$$\begin{gathered} {\mu }_{aX}+b=a_{\mu X}+b \\ {\sigma }_{aX}+b=a_{\sigma X} \end{gathered}$$

Calculation

Adding $0.1$ to every value and then multiplying this by $1.2$

$$\begin{gathered} 1.2(x+0.1)=1.2x+0.12 \\ {\mu }_{1.2X+0.12}=1.2(4.60)+0.12 \\ =5.64 \\ {\sigma }_{1.2X+0.12}=1.2(1.10)=1.32 \end{gathered}$$

Hence, the correct option is (b)