表格显示了 60 株植物的高度 $h$ cm。 计算植物高度标准差的估计值。 答案 ... cm 随机选择两株植物。求这两株植物的高度都大于 30 cm 的概率。 答案 ... 一家牙膏供应商向 3 家不同的商店 A、B 和 C 配送牙膏。每次向每家商店供应的牙膏数量、牙膏的规格和销售价格，以及在 6 个月内向每家商店配送的次数如下表所示。 已知 $T=\begin{pmatrix}68 & 85 & 105\\\*[4pt]75 & 0 & 62\\\*[4pt]0 & 130 & 98\end{pmatrix}.$ 写出矩阵 P，使得 TP 表示每次配送运送到每家商店的牙膏总数量。 答案 P = ... 写出一个 $1 \times 3$ 矩阵 D，表示 6 个月内向每家商店配送的次数。 答案 D = ... 求 DT 的值。 答案 DT = ... 说明 DT 中的元素代表什么。 答案 ... Q 是一个表示每次配送销售所有牙膏后从每家商店获得的总金额的矩阵。
通过矩阵乘法，求 Q。 答案 Q = ...

1. Use midpoints and the grouped-data standard deviation formula

Class midpoints: $15, 25, 35, 45$.

Using a calculator or the grouped-data formula,
$\bar{x}=\frac{7(15)+14(25)+27(35)+12(45)}{60}=32.333\ldots$

$\text{standard deviation}=\sqrt{\frac{7(15-32.333\ldots)^2+14(25-32.333\ldots)^2+27(35-32.333\ldots)^2+12(45-32.333\ldots)^2}{60}}$

$=10.461\ldots\approx 10.5\text{ cm}$

1. Find the number of plants with height greater than $30$ cm

Plants with height greater than $30$ cm are in the classes $30<h\leq 40$ and $40<h\leq 50$.

Number of such plants:
$27+12=39$

2. Calculate the probability that both chosen plants have height greater than $30$ cm

There are $60$ plants in total. Since two plants are chosen without replacement,
$P(\text{both greater than }30)=\frac{39}{60}\times\frac{38}{59}$
$=\frac{13}{20}\times\frac{38}{59}=\frac{247}{590}=\frac{13}{52}$

1. Choose a matrix that sums the three entries in each row of $T$

Since $T$ is a $3\times 3$ matrix and $TP$ must give one total for each shop, $P$ must be a $3\times 1$ column matrix.

To add the numbers of small, medium and large tubes in each row, use
$P=\begin{pmatrix}1\\1\\1\end{pmatrix}.$

1. Write the number of deliveries for shops A, B and C as a row matrix

From the table, the numbers of deliveries over 6 months are:

Shop A: $4$, Shop B: $7$, Shop C: $5$.

Since $D$ is a $1\times3$ matrix,
$D=\begin{pmatrix}4&7&5\end{pmatrix}.$

1. Evaluate the matrix product $DT$

Using $D=\begin{pmatrix}4&7&5\end{pmatrix},\quad T=\begin{pmatrix}68&85&105\\75&0&62\\0&130&98\end{pmatrix},$

$DT=\begin{pmatrix}4&7&5\end{pmatrix}\begin{pmatrix}68&85&105\\75&0&62\\0&130&98\end{pmatrix}$

$=\begin{pmatrix}797&1250&1344\end{pmatrix}.$

1. State the meaning of the elements of $DT$ in context

$D=\begin{pmatrix}4&7&5\end{pmatrix}$ represents the number of deliveries to shops A, B and C over 6 months.

Multiplying by $T$ combines this with the number of tubes per delivery, so
$DT=\begin{pmatrix}797&1250&1344\end{pmatrix}$
represents the total numbers of small, medium and large tubes respectively delivered to all 3 shops over 6 months.

1. Form the price column matrix in the correct order

The sale prices for small, medium and large tubes are $2.50, $4.20 and $6.00 respectively.

$\begin{pmatrix}2.50\\4.20\\6.00\end{pmatrix}$

2. Multiply the toothpaste quantity matrix by the price matrix

$Q=T\begin{pmatrix}2.50\\4.20\\6.00\end{pmatrix}$

$=\begin{pmatrix}68&85&105\\75&0&62\\0&130&98\end{pmatrix}\begin{pmatrix}2.50\\4.20\\6.00\end{pmatrix}$

$=\begin{pmatrix}68(2.50)+85(4.20)+105(6.00)\\75(2.50)+0(4.20)+62(6.00)\\0(2.50)+130(4.20)+98(6.00)\end{pmatrix}$

$=\begin{pmatrix}1166\\559.50\\1134\end{pmatrix}$