Question

a student was wondering if movies that cost more money to make will have better reviews. they collected the budget and rotten tomato scores of fifteen movies (shown below). help them by creating a scatter plot and finding a line of best fit.
budget (in 100 millions) 5.7 5.7 5.8 5.9 6.1
rotten tomato score 54 42 96 89 55
budget (in 100 millions) 6.2 6.2 6.3 6.4 6.6
rotten tomato score 54 96 97 95 90
budget (in 100 millions) 6.6 6.9 7.0 7.8 10.2
rotten tomato score 69 95 84 85 98

Explanation:

Step1: Recall the formula for the line of best - fit in simple linear regression.

The line of best - fit for a simple linear regression is of the form $y = mx + b$, where $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $b=\bar{y}-m\bar{x}$. Let $x$ be the budget (in 100 millions) and $y$ be the Rotten Tomato score. First, calculate the necessary sums:
Let $n = 15$.
Calculate $\sum_{i = 1}^{n}x_i$, $\sum_{i = 1}^{n}y_i$, $\sum_{i = 1}^{n}x_i^{2}$, $\sum_{i = 1}^{n}x_iy_i$.
$\sum_{i = 1}^{15}x_i=5.7 + 5.8+5.9+6.1+6.2+6.2+6.3+6.4+6.6+6.6+6.9+7.0+7.8+10.2$
$=94.7$
$\sum_{i = 1}^{15}y_i=42 + 54+55+89+96+54+90+95+97+96+54+69+84+95+98$
$=1124$
$\sum_{i = 1}^{15}x_i^{2}=5.7^{2}+5.8^{2}+5.9^{2}+6.1^{2}+6.2^{2}+6.2^{2}+6.3^{2}+6.4^{2}+6.6^{2}+6.6^{2}+6.9^{2}+7.0^{2}+7.8^{2}+10.2^{2}$
$=607.97$
$\sum_{i = 1}^{15}x_iy_i=5.7\times42+5.8\times54+5.9\times55+6.1\times89+6.2\times96+6.2\times54+6.3\times90+6.4\times95+6.6\times97+6.6\times96+6.9\times54+7.0\times69+7.8\times84+10.2\times95$
$=7377.3$

Step2: Calculate the slope $m$.

$m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$
$=\frac{15\times7377.3 - 94.7\times1124}{15\times607.97-(94.7)^{2}}$
$=\frac{110659.5-106442.8}{9119.55 - 8968.09}$
$=\frac{4216.7}{151.46}\approx27.84$

Step3: Calculate the mean of $x$ and $y$.

$\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{94.7}{15}\approx6.313$
$\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{1124}{15}\approx74.933$

Step4: Calculate the y - intercept $b$.

$b=\bar{y}-m\bar{x}$
$b = 74.933-27.84\times6.313$
$b=74.933 - 175.764$
$b\approx - 100.83$

The line of best - fit is $y = 27.84x-100.83$

Answer:

$y = 27.84x - 100.83$