The table shows the height, h cm, of 60 plants. Calculate an estimate of the standard deviation of the height of the plants. Answer ... cm Two plants are chosen at random. Find the probability that both plants have height greater than 30 cm. Answer ... A toothpaste supplier delivers tubes of toothpaste to 3 different shops A, B and C. The number of tubes of toothpaste supplied per delivery to each shop, the sizes and sale prices of the tubes, together with the number of deliveries made to each shop over a 6-month period are shown in the table below. It is given that $T=\begin{pmatrix}68 & 85 & 105\\[4pt]75 & 0 & 62\\[4pt]0 & 130 & 98\end{pmatrix}.$ Write down the matrix P such that TP gives the total number of tubes of toothpaste delivered to each shop per delivery. Answer P = ... Write down a 1 × 3 matrix D to represent the number of deliveries over 6 months to each shop. Answer D = ... Evaluate DT. Answer DT = ... State what the elements of DT represent. Answer ... Q is a matrix that represents the total amount of money that would be obtained from each shop from the sale of all the tubes of toothpaste per delivery.
By matrix multiplication, find Q. Answer 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

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

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

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

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