Answer:
a) y = -441.105 + 0.2232x
b) 2043
Step-by-step explanation:
Correlation measures how closely two variables are linked.
If two variables are correlated, you can draw a line of best fit on the scatter plot.
- Explanatory (independent) variable is drawn on the x-axis.
- Response (dependent) variable is drawn on the y-axis.
Plot the points from the table where:
- x = year
- y = average price (in dollars)
(See attached graph).
We can see that the points lie close to a straight line. Therefore, a line of best fit with a positive slope can be drawn, since the two variables are positively correlated.
Linear regression is a method for finding the equation of a line of best fit on a scatter plot.
The regression line of y on x:
[tex]y = a + bx[/tex]
where a = y-intercept and b = slope
- If b is positive, the variables are positively correlated.
- If b is negative, the variables are negatively correlated.
We cannot find the y-intercept from inspection of the scatter plot (attached) since we would need to extend the x-axis to thousands of years.
Therefore, to find the regression equation, use a graphing calculator:
[tex]y=-441.105+0.2232x[/tex]
To find the approximate year the movie ticket price will reach $15, substitute y = 15 into the equation and solve for x:
[tex]\implies -441.105+0.2232x=15[/tex]
[tex]\implies 0.2232x=456.105[/tex]
[tex]\implies x=2043.481...[/tex]
Therefore, the approximate year the movie ticket price will reach $15 is during 2043.
Note:
For this problem, we have used the line of best fit (regression line) to predict a value of that is outside the range of the original data. When using values outside the range to predict corresponding values, this is called extrapolation. These predictions can be unreliable because there is no evidence that the relationship described by the regression line is true for all values of x. Therefore, caution should be applied in these cases.
