A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the table.

Question

High temperature (°F)	tNumber of cans sold	N
55	345
58	330
64	410
68	455
70	445
75	605
80	735
84	780

 Find a linear function that models the data. (Round your answers to two decimal places.)

N=

Thanks!

Answer 1

So, it appears that the store manager has noticed a relationship between the temperature and sales of soft drinks. In particular, he believes that as the temperature increases, the sales increase as well. So, we can predict sales with temperature. That is, our response (Y) variable is sales and our predictor (X) variable is temperature. We would like to model the relationship between X and Y using the form Y = f(X) + error. (Response = Prediction + Error.) We only have a single predictor.

Let the intercept parameter be beta_0 and let our slope parameter be beta_1. Then, our linear model is

Y_i = beta_0 + beta_1 * x_i + some_error

Don't worry about the "some_error" term. You just need to calculate the beta_0 and beta_1 parameters using information below.

x_bar = avg(X)

y_bar = avg(Y)

beta_0 = y_bar - beta_1 * x_bar

beta_1 = [ SUM from 1 to n of ( (x_i - x_bar) * (y_i - y_bar) ) ] / [ SUM from 1 to n of (x_i - x_bar)^2 ]

Can you take it from here?

Answer 2

In order to determine the answer to this problem, you'll need to find the line of best fit.

Remember that a linear function is best explained by y =(m)x + b where m is the slope and b is the y intercept.

There are two ways to determine the line of best fit.

1) Plot the points on a map and then draw a line that best fits the data. While this method works, it is not the most accurate method due to human error.

2) Use a calculator or a graphing function to find the line of best fit.

1. Traditionally, I use a graphing calculator to determine this. However, for ease of use, I'll use the following website: http://www.endmemo.com/statistics/lr.php
2. Enter the data points as described in the above data
3. X = high temperature (independent variable)
4. Y = number of cans sold (dependent variable)
5. Determine the data:
6. Intercept = -619.76
7. Slope = 16.36
8. Write the line of best fit
9. y = 16.36x - 619.76