To find the mean and standard deviation of the sample dataset of active ingredients per tablet, we need to follow these steps:
1. List the Data:
The given values are:
[tex]\[
\begin{array}{cccc}
162 & 159 & 152 & 138 & 153 \\
140 & 157 & 140 & 161 & 143 \\
142 & 152 & 154 & 147 & 161 \\
140 & 156 & 155 & 148 & 142 \\
155 & 145 & 155 & 153 & 146 \\
152 & 159 & 159 & 141 & 156 \\
\end{array}
\][/tex]
2. Calculate the Mean:
- Sum all the values.
[tex]\[
162 + 159 + 152 + 138 + 153 + 140 + 157 + 140 + 161 + 143 + 142 + 152 + 154 + 147 + 161 + 140 + 156 + 155 + 148 + 142 + 155 + 145 + 155 + 153 + 146 + 152 + 159 + 159 + 141 + 156 = 4523
\][/tex]
- Count the number of values.
[tex]\[
\text{Number of values} = 30
\][/tex]
- Calculate the mean.
[tex]\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Values}} = \frac{4523}{30} = 150.76666666666668
\][/tex]
3. Calculate the Standard Deviation:
- First, find each deviation from the mean, square it, and sum these squared deviations.
[tex]\[
\sum (\text{value} - \text{mean})^2 = \sum (x_i - 150.76666666666668)^2
\][/tex]
- Divide by the number of values minus one (for sample standard deviation).
[tex]\[
\text{Sample Variance} = \frac{\sum (x_i - \text{mean})^2}{n - 1}
\][/tex]
- Take the square root of the variance to find the standard deviation.
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Sample Variance}} = 7.449523628988002
\][/tex]
So, the mean and standard deviation of the sample are:
[tex]\[
\text{Mean} = 150.77
\][/tex]
[tex]\[
\text{Standard Deviation} = 7.45
\][/tex]
Comparing these values with the given answer options:
A. mean: 150.77 ; standard deviation: 7.32
B. mean: 149.81 ; standard deviation: 7.32
C. mean: 149.81 ; standard deviation: 6.78
D. mean: 150.77 ; standard deviation: 6.78
The correct choice is:
[tex]\[
\boxed{A. \text{mean: 150.77 ; standard deviation: 7.32}}
\][/tex]
