Exercise 1 Page 680

We are given a table that shows the life expectancy, in years, for people born in certain years.

Years Since $1900$	$0$	$10$	$20$	$30$	$40$	$50$	$60$	$70$	$80$	$90$	$100$
Life Expectancy	$47.3$	$50.0$	$54.1$	$59.7$	$62.9$	$68.2$	$69.7$	$70.8$	$73.7$	$75.4$	$77.1$

We are asked to construct a scatter plot and draw a line of best fit. To construct a scatter plot, we need to sketch each point from the table on a coordinate plane. The horizontal axis will represent the years since $1900$ and the vertical axis will represent the life expectancy of the people born.

For example, let's sketch the third point from the table $(20, 54.1) .$

In a similar way, we will now graph the other points from the table.

We constructed a scatter plot of the data! Now we will draw a straight line that best represents the data. Remember that we should try for the line to be as close to the data points as possible.

The points are very close to the line and a similar number of points is above and below the line. Therefore, this is a line of best fit. Notice that, if you sketch a different line that is close to the data points, it would also be a correct answer.

In Part A, we drew the following line of best fit.

Let's write an equation in slope-intercept form for this line!

y = m x + b

In this equation,

m

is the slope and

b

is the

y -

intercept. First, we will choose any two points on the line.

The first point has coordinates

x_{1} = 70

and

y_{1} = 70.8 .

The second point has coordinates

x_{2} = 80

and

y_{2} = 73.7 .

Let's substitute these values into slope formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstituteValues

Substitute values

m = \frac{7 3 . 7 - 7 0 . 8}{8 0 - 7 0}

SubTerms

Subtract terms

m = \frac{2 . 9}{1 0}

CalcQuot

Calculate quotient

m = 0.29

We found that the slope

m

is equal to

0.29 .

This means that the life expectancy of a person born in a certain year is longer by

0.29

years than the life expectancy of a person born in a previous year.

y = m x + b \Leftrightarrow y = 0.29 + b

The

y -

intercept

b

is the

y -

value when

x = 0 .

We can find it on the graph. Let's do it!

We can see that the

y -

intercept is equal to about

50 .

This means that in the year

1900,

the life expectancy was approximately equal to

50

years. Finally, we can write a complete equation in slope-intercept form for our line of fit.

y = 0.29 x + b ⇕ y = 0.29 x + 50

In Part B, we wrote an equation that estimates the life expectancy

y

for people born

x

years after the year

1900 .

y = 0.29 x + 50

Here, we are asked to use this equation to make a conjecture about the life expectancy for a person born in

2020 .

To do that, we first need to calculate the number of years

x

between the year

1900

and

2020 .

x = 2020 - 1900 \Leftrightarrow x = 120

Now we can substitute

x = 120

into the equation from Part B to estimate the life expectancy for a person born in

2020 .

y = 0.29 x + 50

Substitute

$x = 120$

y = 0.29 (120) + 50

Multiply

y = 34.8 + 50

AddTerms

Add terms

y = 84.8

Using the equation, we estimated that the life expectancy of a person born in

2020

84.8

years.

Subchapter links

Guided Practice p.680

Independent Practice p.681-682

H.O.T. Problems p.682

Standardized Test Practice p.682-684

Extra Practice p.683

Common Core Review p.684