Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Central Tendency, Exercise 7: EXERCISE 2D

The wind speeds, $x$ miles per hour $\left(\mathrm{mph}\right)$, were measured at a coastal location at midday on $40$ consecutive days and are presented in the following table.

Abel wishes to calculate an estimate of the mean wind speed in kilometres per hour $\left(\mathrm{km}/\mathrm{h}\right).$ He knows that a distance of $5$ miles is approximately equal to $8 \mathrm{km}.$

Use the wind speeds in mph to estimate the mean wind speed in $\mathrm{km}/\mathrm{h}.$

Given that $15$ values of $x$ are such that $\Sigma \left(3x-2\right)=528,$ find $\overline{x}$ and find the value of $b$ such that $\Sigma \left(0.5x-b\right)=138.$

For $20$ values of $y,$ it is given that $\Sigma \left(ax-b\right)=400$ and $\Sigma \left(bx-a\right)=545.$ Given also that $\overline{x}=6.25,$ find the value of $a$ and of $b.$

The midpoint of the line segment between $A$ and $B$ is at $\left(5.2, -1.2\right).$

Find the coordinates of the midpoint after the following transformations have been applied to $A$ and to $B.$

$T:$ Translation by the vector $\left(\begin{array}{r}-7\\ 4\end{array}\right).$

The midpoint of the line segment between $A$ and $B$ is at $\left(5.2, -1.2\right).$

Find the coordinates of the midpoint after the following transformations have been applied to $A$ and to $B.$

$E:$ Enlargement through the origin with scale factor $5.$

The midpoint of the line segment between $A$ and $B$ is at $\left(5.2, -1.2\right).$

Find the coordinates of the midpoint after the following transformations have been applied to $A$ and to $B.$

Transformations $T$ and $E$ are carried out one after the other. Investigate whether the location of the midpoint of $AB$ is independent of the order in which the transformations are carried out.

Five investors are repaid, each with their initial investment increased by $p\%$ plus a fixed 'thank you' bonus of $\$q.$ The woman who invested $\$20000$ is repaid double her investment and the man who invested $\$7500$ is repaid triple his investment. Find the total amount that the five people invested, given that the mean amount repaid to them was $\$33000.$

Do you think the method of repayment is fair? Give a reason for your answer.

One of the units used to measure pressure is pounds per square inch (psi). The mean pressure in the four tyres of a particular vehicle is denoted by $\overline{x}$ psi. Given that $1$ pound is approximately equal to $0.4536 \mathrm{kg}$ and that $1$ metre is approximately equal to $39.37$ inches, express the sum of the pressures in the four tyres of this vehicle in grams per ${\mathrm{cm}}^2.$