Question

1. the table below shows the average price of a movie ticket during certain years.

year	price
2003	6.03
2004	6.21
2005	6.41
2006	6.55
2007	6.88
2008	7.18
2009	7.35

a) find the regression equation.
b) during what approximate year will the price of a movie ticket reach $15?

Explanation:

Step1: Let $x$ be the number of years since 2002.

Let $x = 0$ for year 2002, $x = 1$ for year 2003 and so on. The data - points are $(0,5.80),(1,6.03),(2,6.21),(3,6.41),(4,6.55),(5,6.88),(6,7.18),(7,7.35)$.

Step2: Calculate the mean of $x$ and $y$.

The mean of $x$ values $\bar{x}=\frac{0 + 1+2+3+4+5+6+7}{8}=\frac{28}{8}=3.5$.
The mean of $y$ values $\bar{y}=\frac{5.80 + 6.03+6.21+6.41+6.55+6.88+7.18+7.35}{8}=\frac{52.41}{8}=6.55125$.

Step3: Calculate the slope $m$.

$m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$.
$\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=(0 - 3.5)(5.80 - 6.55125)+(1 - 3.5)(6.03 - 6.55125)+(2 - 3.5)(6.21 - 6.55125)+(3 - 3.5)(6.41 - 6.55125)+(4 - 3.5)(6.55 - 6.55125)+(5 - 3.5)(6.88 - 6.55125)+(6 - 3.5)(7.18 - 6.55125)+(7 - 3.5)(7.35 - 6.55125)$
$=(-3.5)(-0.75125)+(-2.5)(-0.52125)+(-1.5)(-0.34125)+(-0.5)(-0.14125)+(0.5)(-0.00125)+(1.5)(0.32875)+(2.5)(0.62875)+(3.5)(0.79875)$
$=2.629375 + 1.303125+0.511875 + 0.070625-0.000625+0.493125+1.571875+2.795625$
$=9.374$.
$\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=(0 - 3.5)^{2}+(1 - 3.5)^{2}+(2 - 3.5)^{2}+(3 - 3.5)^{2}+(4 - 3.5)^{2}+(5 - 3.5)^{2}+(6 - 3.5)^{2}+(7 - 3.5)^{2}$
$=12.25+6.25 + 2.25+0.25+0.25+2.25+6.25+12.25$
$=42$.
So, $m=\frac{9.374}{42}\approx0.2232$.

Step4: Calculate the $y$ - intercept $b$.

$b=\bar{y}-m\bar{x}=6.55125-0.2232\times3.5=6.55125 - 0.7812=5.77$.
The regression equation is $y = 0.2232x+5.77$.

Step5: Solve part (b).

Set $y = 15$ in the regression equation $15=0.2232x+5.77$.
$0.2232x=15 - 5.77=9.23$.
$x=\frac{9.23}{0.2232}\approx41.35$.
Since $x$ is the number of years since 2002, the year is approximately $2002 + 41=2043$.

Answer:

a) The regression equation is $y = 0.2232x+5.77$.
b) Approximately in the year 2043.