The shipment times in hours for a load of goods from a factory to market are tabulated in the following frequency distribution. Find the sample standard deviation of shipment time. $$\begin{array} { | l | l | l | l | }
\hline \text { Shipment time (hours) } & \text { Midpoint } x & \text { Frequency } f & \text { Product } x f \
\hline 22.5 - 28.5 & & 1 & \
\hline 28.5 - 34.5 & & 37 & \
\hline 34.5 - 40.5 & & 82 & \
\hline 40.5 - 46.5 & & 28 & \
\hline 46.5 - 52.5 & & 13 & \
\hline 52.5 - 58.5 & & 4 & \
\hline
\end{array}$$ Round your answer to the nearest tenth.
A) 3.5 h
B) 5.5 h
C) 6.5 h
D) 4.5 h
E) 5.8 h

Question

The shipment times in hours for a load of goods from a factory to market are tabulated in the following frequency distribution. Find the sample standard deviation of shipment time. $$\begin{array} { | l | l | l | l | } 
\hline \text { Shipment time (hours) } & \text { Midpoint } x & \text { Frequency } f & \text { Product } x f \
\hline 22.5 - 28.5 & & 1 & \
\hline 28.5 - 34.5 & & 37 & \
\hline 34.5 - 40.5 & & 82 & \
\hline 40.5 - 46.5 & & 28 & \
\hline 46.5 - 52.5 & & 13 & \
\hline 52.5 - 58.5 & & 4 & \
\hline
\end{array}$$ Round your answer to the nearest tenth.
A) 3.5 h
B) 5.5 h
C) 6.5 h
D) 4.5 h
E) 5.8 h

Quizplus · Accepted Answer

To calculate the sample standard deviation, we first need to find the midpoint for each class, multiply each midpoint by its frequency to get the $xf$ column, and then use these values to find the mean and the sum of squared deviations from the mean.1. Calculate the midpoints ($x$) for each class:   - $22.5 - 28.5$: $25.5$   - $28.5 - 34.5$: $31.5$   - $34.5 - 40.5$: $37.5$   - $40.5 - 46.5$: $43.5$   - $46.5 - 52.5$: $49.5$   - $52.5 - 58.5$: $55.5$2. Multiply each midpoint by its frequency ($f$) to get the $xf$ column:   - $25.5 \times 1 = 25.5$   - $31.5 \times 37 = 1165.5$   - $37.5 \times 82 = 3075$   - $43.5 \times 28 = 1218$   - $49.5 \times 13 = 643.5$   - $55.5 \times 4 = 222$3. Calculate the sum of $f$ and the sum of $xf$:   - Sum of $f = 1 + 37 + 82 + 28 + 13 + 4 = 165$   - Sum of $xf = 25.5 + 1165.5 + 3075 + 1218 + 643.5 + 222 = 6350$4. Find the mean ($\bar{x}$):   - $\bar{x} = \frac{\text{Sum of } xf}{\text{Sum of } f} = \frac{6350}{165} \approx 38.5$5. Calculate the squared deviations from the mean ($(x - \bar{x})^2$) and multiply by frequency ($f$) for each class, then sum these values:   - This involves calculating $(x - \bar{x})^2f$ for each class and summing. For brevity, the detailed calculation is not shown, but the sum of these values is crucial for the next step.6. Calculate the sample variance ($s^2$):   - $s^2 = \frac{\text{Sum of } (x - \bar{x})^2f}{n - 1}$, where $n$ is the sample size (165 in this case).7. Finally, find the sample standard deviation ($s$) by taking the square root of the sample variance.Given the data and the process, the correct calculation leads to a sample standard deviation that rounds to approximately 5.8 hours, making choice E the correct answer.

The Shipment Times in Hours for a Load of Goods