下表显示了 X 厂制造 of 50 辆汽车每升汽油行驶的距离。

1. Find the class midpoints and calculate the estimated mean using the grouped-data mean formula.

Class midpoints: $20.5,\ 21.5,\ 22.5,\ 23.5$

$\text{Estimated mean}=\frac{20.5(9)+21.5(13)+22.5(17)+23.5(11)}{50}$

$=\frac{184.5+279.5+382.5+258.5}{50}$

$=\frac{1105}{50}=22.1$

1. Use the grouped-data standard deviation formula with the class midpoints.

Class midpoints: $20.5, 21.5, 22.5, 23.5$

From part (a), estimated mean $\bar{x}=22.1$.

$$\begin{aligned} \sum fx^2&=9(20.5)^2+13(21.5)^2+17(22.5)^2+11(23.5)^2\\ &=24472.5 \end{aligned}$$ $$\begin{aligned} \text{standard deviation} &=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}^2}\\ &=\sqrt{\frac{24472.5}{50}-(22.1)^2}\\ &=\sqrt{489.45-488.41}\\ &=\sqrt{1.04}\\ &=1.0198\ldots\\ &\approx 1.02 \end{aligned}$$

1. Compare the estimated means of Factory X and Factory Y.

For Factory X, estimated mean $=22.1$ km.

For Factory Y, estimated mean $=22.5$ km.

Since $22.5>22.1$, cars manufactured by Factory Y travel a greater distance per litre on average.

2. Compare the standard deviations of Factory X and Factory Y.

For Factory X, standard deviation $=1.02$ km.

For Factory Y, standard deviation $=0.985$ km.

Since $0.985<1.02$, the distances for Factory Y are less spread out, so Factory Y cars are more consistent in distance travelled per litre.

1. Decide how adding the new 5 cars affects the mean.

The original estimated mean for Factory Y is $22.5$ km.

The 5 additional cars travelled between $22$ km and $23$ km, so using the same grouped-data method, their representative value is the class midpoint

$\frac{22+23}{2}=22.5.$

Since the added cars have the same estimated value as the original mean, the mean remains unchanged.

1. Decide the effect of adding 5 cars whose estimated value is at the existing mean.

The new 5 cars are in the $22$ km to $23$ km class, so their grouped-data estimated value is the midpoint:

$\frac{22+23}{2}=22.5\text{ km}$

This is equal to the existing mean of Factory Y, $22.5$ km. Adding values at the mean adds no extra spread, so the standard deviation will decrease.