10.3 Homework
Questic
A sample of 32 households from a survey generates the following frequency table of household income (in thousands). Find the standard deviation.
\begin{tabular}{c|c}
Interval & Frequency \\
\hline $0-29$ & 9 \\
$30-59$ & 5 \\
$60-89$ & 10 \\
$90-119$ & 4 \\
$120-149$ & 0 \\
$150-179$ & 1 \\
$180-209$ & 3
\end{tabular}
$s=54.3$ thousand dollars
(Round to one decimal place as needed)

Step 1 ：Given a frequency table of household income, we are asked to find the standard deviation. The frequency table is as follows: \[\begin{tabular}{c|c} Interval & Frequency \\ \hline $0-29$ & 9 \\ $30-59$ & 5 \\ $60-89$ & 10 \\ $90-119$ & 4 \\ $120-149$ & 0 \\ $150-179$ & 1 \\ $180-209$ & 3 \end{tabular}\]

Step 2 ：The first step is to find the midpoint of each interval. The midpoints are: \[14.5, 44.5, 74.5, 104.5, 134.5, 164.5, 194.5\]

Step 3 ：Next, we multiply each midpoint by its corresponding frequency and sum these products to find the total. The total frequency is 32.

Step 4 ：We then divide the total by the total frequency to find the mean. The mean is \(70.75\) thousand dollars.

Step 5 ：We can now use the formula for the standard deviation of a frequency distribution to find the standard deviation. The standard deviation is approximately \(53.42927568290628\) thousand dollars.

Step 6 ：Finally, we round the standard deviation to one decimal place. The standard deviation of the household income is \(\boxed{53.4}\) thousand dollars.