Answer
The mean and the standard deviation of the tax rate of cigarettes are 1.505 and 0.982.
The approximate mean and standard deviation are nearly equal with the actual mean and standard deviation.
The following steps are followed to obtain the mean and the standard deviation for the provided data:
Step 1: Obtain the midpoint\[\left( {{x}_{i}} \right)\] of the class interval by adding the consecutive two lowerclass boundaries and divide the sum by 2.
Step 2: The formula used to obtain the mean\[\left( {\bar{x}} \right)\]is:
Work Step by Step
Therefore, the mean tax rates of cigarettes is:
\[\begin{align}
& \bar{x}=\frac{\sum{{{x}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}} \\
& =\frac{76.75}{51} \\
& =1.505
\end{align}\]
And the standard deviation is calculated as:
\[\begin{align}
& s=\sqrt{\frac{\sum{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}{{f}_{i}}}}{\sum{{{f}_{i}}}-1}} \\
& =\sqrt{\frac{49.19}{51-1}} \\
& =0.982
\end{align}\]
The actual mean is calculate as:
\[\begin{align}
& \bar{x}=\frac{\sum{{{x}_{i}}}}{n} \\
& =\frac{3.03+1.18+\cdots +3.00}{51} \\
& =1.45
\end{align}\]
The actual standard deviation is calculated as:
\[\begin{align}
& \sigma =\sqrt{\frac{1}{n}\sum{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}} \\
& =\sqrt{\frac{1}{51}\left( {{\left( 3.03-1.45 \right)}^{2}}+{{\left( 1.18-1.45 \right)}^{2}}+\cdots +{{\left( 3.00-1.45 \right)}^{2}} \right)} \\
& =0.93
\end{align}\]
Therefore, the actual mean and standard deviation of the tax rates is nearly equal to the approximate mean and standard deviation, respectively.
