Problem

The parking times, $t$ hours, for cars in a car park are summarised below.

Time ($t$ hours)	Frequency ($f$)	Time midpoint ($m$)
$0\leqslant t<1$	10	0.5
$1\leqslant t<2$	18	1.5
$2\leqslant t<4$	15	3
$4\leqslant t<6$	12	5
$6\leqslant t<12$	5	9

(You may use $\sum fm=182$ and $\sum fm^2=883$)

A histogram is drawn to represent these data.

The bar representing the time $1\leqslant t<2$ has a width of 1.5 cm and a height of 6 cm.

(a) Calculate the width and the height of the bar representing the time $4\leqslant t<6$

(b) Use linear interpolation to estimate the median parking time for the cars in the car park.

(c) Estimate the mean and the standard deviation of the parking time for the cars in the car park.

(d) Describe, giving a reason, the skewness of the data.

One of these cars is selected at random.

(e) Estimate the probability that this car is parked for more than 75 minutes.

中文翻译

某停车场内汽车的停车时间 $t$（小时）如表所示。（可使用 $\sum fm=182$ 和 $\sum fm^2=883$。）

现根据这些数据绘制直方图。区间 $1\leqslant t<2$ 的长条宽 1.5 cm、高 6 cm。

(a) 计算区间 $4\leqslant t<6$ 的长条宽度及高度。

(b) 用线性插值估计停车时间的中位数。

(c) 估计停车时间的平均数及标准差。

(d) 描述数据的偏态，并说明理由。

随机选取其中一辆汽车。

(e) 估计该车停车超过 75 分钟的概率。