Question

the table shows the number of tickets sold for a concert when different prices are charged. write an equation of a line of fit for the data. round all values to the nearest tenth.

ticket price (dollars), x	17	20	22	26
tickets sold, y	450	423	400	395

$y=square x+square$

does it seem reasonable to use your model to predict the number of tickets sold when the ticket price is $85? explain.

yes; predictions can always be made using a line of fit.

yes; the data show a strong negative correlation for the line of fit.

no; the value 85 is not close to the values used to create the line of fit.

no; predictions can only be made for values used to create the line of fit.

Explanation:

Step1: Calculate the slope

We use the formula for the slope $m=\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}$, where $n = 4$.
$\sum x=17 + 20+22 + 26=85$, $\sum y=450+423 + 400+395 = 1668$, $\sum xy=17\times450+20\times423+22\times400+26\times395=7650+8460+8800+10270 = 35180$, $\sum x^{2}=17^{2}+20^{2}+22^{2}+26^{2}=289+400+484+676 = 1849$.
$m=\frac{4\times35180 - 85\times1668}{4\times1849-85^{2}}=\frac{140720-141780}{7396 - 7225}=\frac{-1060}{171}\approx - 6.2$.

Step2: Calculate the y - intercept

We use the formula $b=\overline{y}-m\overline{x}$, where $\overline{x}=\frac{\sum x}{n}=\frac{85}{4}=21.25$ and $\overline{y}=\frac{\sum y}{n}=\frac{1668}{4}=417$.
$b = 417-(-6.2)\times21.25=417 + 131.75=548.8$.

(for second part):
The values of $x$ used to create the line of fit are 17, 20, 22 and 26. The value $x = 85$ is far from these values. Extrapolating this far from the data points can lead to inaccurate predictions.

Answer:

$y=-6.2x + 548.8$