Question

Find the regression equation, letting the first variable be the predictor (x) variable. Using the listed actress/actor ages in various years, find the best predicted age of the Best Actor winner given that the age of the Best Actress winner that year is 62 years. Is the result within 5 years of the actual Best Actor winner, whose age was 45 years?

Col1 Best Actress 29 30 30 62 33 34 43 30 65 23 46 54

Col2 Best Actor 43 36 40 45 52 49 63 48 39 53 46 35

Find the equation of the regression line (Round the constant to one decimal place as needed. Round the coefficient to three decimal places as needed.) The best predicted age ofthe Best Actor winner given that the age ofthe Best Actres s winner that year is 62 years is□years old Round to the nearest whole number as needed.) Is the result within 5 years of the actual Best Actor winner, whose age was 45 years?

Answer 1

Answer:

The equation of the regression line is: $y~=~51.5 ~-~ 0.143 \cdot x$

The best predicted age of the Best Actor winner given that the age of the Best Actress winner that year is 62 years is 43 years.

The actual age of the Best Actor is given as 45 years. Thus the predicated age is within 5 years of the actual age.

Step-by-step explanation:

Linear regression is a linear approach to modelling the relationship between a dependent variable and one or more independent variables.

Let X be the independent variable and Y be the dependent variable. We will define a linear relationship between these two variables as follows:

$Y=bX+a$

We have the following data:

$\left\begin{array}{cc}\mathrm{Best \:Actress}&\mathrm{Best \: Actor}\\29&43\\30&36\\30&40\\62&45\\33&52\\34&49\\43&63\\30&48\\65&39\\23&53\\46&46\\54&35\end{array}\right$

To find the line of best fit for the points, follow these steps:

Step 1: Find $X\cdot Y$ and $X\cdot X$ as it was done in the below table.

Step 2: Find the sum of every column:

$\sum{X} = 479 ~,~ \sum{Y} = 549 ~,~ \sum{X \cdot Y} = 21608 ~,~ \sum{X^2} = 21265$

Step 3: Use the following equations to find intercept a and slope b:

$\begin{aligned}        a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} =             \frac{ 549 \cdot 21265 - 479 \cdot 21608}{ 12 \cdot 21265 - 479^2} \approx 51.5 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}        = \frac{ 12 \cdot 21608 - 479 \cdot 549 }{ 12 \cdot 21265 - \left( 479 \right)^2} \approx -0.143\end{aligned}$

Step 4: Assemble the equation of a line

$\begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~51.5 ~-~ 0.143 \cdot x\end{aligned}$

The best predicted age of the Best Actor winner given that the age of the Best Actress winner that year is 62 years is

$y~&=~51.5 ~-~ 0.143 \cdot 62\\y~&=43$

The actual age of the Best Actor is given as 45 years. Thus the predicated age is within 5 years of the actual age.