Exercise 11 Page 211

Practice makes perfect

a Let's consider the given data!

Advertising (dollars), x	Yearly attendance, y
500	400
1000	550
1500	550
2000	800
2500	650
3000	800
3500	1050
4000	1100

We are asked to find an equation of the line of best fit for the above data. We will begin by entering the values into list. We press the button STAT on the calculator. After that we choose Edit and then enter the x-values in column L1 and the y-values in column L2.

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

To view the linear regression analysis of the dataset we press STAT, scroll to right to view the CALC options, and then choose the fourth option in the list, LinReg(ax+b).

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

We find the equation of the line of best fit by substituting the values for a and b into the slope-intercept form equation. y= ax+ b ⇓ y= 0.19x+ 309

b The correlation coefficient is the value stated as r on the screen of the calculator.

For our case, the correlation coefficient is approximately 0.94. Since it is close to 1, there is a strong correlation between the advertising and the yearly attendance.

c As we stated in Part B, there is strong correlation between the variables in the given data.

r≈ 0.94 ⇒ Strong correlation Therefore, we should expect that the points in the scatter plot of the residuals will be close to and evenly dispersed about the horizontal axis.

d Let's recall what it means that two variables have a causal relationship.

A causal relationship exists when one variable causes a change in another variable.

If the advertising is increased, more people learn about the festival and that will make more people visit. Therefore, there is a causal relationship between the variables.

e Notice that the value we are asked to predict is outside the range of known values. Therefore, we should use the equation of the line of best fit we found in Part A. y=0.19x+309 We will substitute 2000 for y into the equation and solve it for x. Let's do it!

y=0.19x+309

Substitute

y= 2000

2000=0.19x+309

▼

Solve for x

SubEqn

LHS-309=RHS-309

1691=0.19x

DivEqn

.LHS /0.19.=.RHS /0.19.

8900=x