Exercise 17 Page 197

Practice makes perfect

a We have been given a table with data for

x

and

y .

Minutes, $x$	People, $y$
$1$	$10$
$2$	$100$
$3$	$400$
$4$	$900$
$5$	$1400$
$6$	$1800$
$7$	$2100$

In order to find a line of fit using our calculator, we need to first enter the values. Let's press the $STAT$ button.

Then we choose the first option in the menu, Edit, and fill in the values in lists L1 and L2.

Illustration of the lists on the calculator with six ordered pairs written

We can perform a regression analysis on the data by pressing the $STAT$ button again, followed by using the right-arrow key to select the CALC menu.

This menu lists the various regressions that are available. If we choose the fourth option in the menu LinReg(ax+b) and press $ENTER,$ the calculator performs a linear regression using the data that was entered.

Illustration of the LinReg(ax+b) window on the calculator

We will substitute the values of

a

and

b

into the equation

y = a x + b

to find the equation of the line of best fit.

y = 381 x - 566

Next we want to graph the data and the line of fit on the same coordinate plane. Let's press

Y =

on the calculator. Then we write the equation on one of the rows.

Illustration of the Y= window on the calculator.

To adjust the viewing window we press the button $WINDOW$ and set the values to fit our data.

Illustration of the WINDOW option, how to set the size of the viewing window, on the calculator.

To make the calculator graph both the line and the scatter plot we need to press $STAT PLOT .$

If the scatter plot is Off or if we want to change any settings we press $ENTER .$

To activate the scatter plot place the cursor on the option On and press $ENTER .$ We need to select the first symbol in Type to get a scatter plot. We can now draw the scatter plot and the line of fit by pressing $GRAPH .$

Scatter plot of the data with line of fit as drawn on a calculator

b The correlation coefficient is represented with the variable

r .

Let's consider the linear regression output.

We find the correlation coefficient in the last row.

r = 0.989

The correlation coefficient

r

is always between

- 1 \leq r \leq 1,

where positive values represent a positive slope and negative values represent a negative slope. Additionally, the closer the value is to

0,

the weaker the correlation. Our value,

r = 0.989,

tells us that the correlation is positive and strong.

c In Part A, we found the equation for the line of best fit.

y = 381 x - 566

We know that the slope is

381

and the

y -

intercept is

- 566 .

To interpret these values we need to understand that

x

is the number of minutes that have passed and

y

is the number of people who reported feeling the earthquake. Recall that slope is the

change

in

y

divided by the

change

in

x .

Slope \frac{Δ y}{Δ x} = 381 ⇕ = \frac{3 8 1 people}{1 minute}

This means that on average

381

people reported the earthquake each minute after it hit. The

y -

intercept does not actually make any sense in the context of the problem, as there cannot be a negative number of people reporting the earthquake. It is an extension of the line beyond reasonable boundaries.

Subchapter links

Communicate Your Answer p.191

Monitoring Progress p.193-195

Exercises p.196-198

Exercise 17 Page 197

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