Any help for this question please?

Question

The following table shows the hours of sunshine, x, during nine days in August and the number of ice creams, y, sold by a beach shop in Cornwall.

x	4.3	6.9	0.0	10.4	5.2	1.8	8.0	9.2	2.1
y	224	208	123	419	230	184	362	351	196

 (a)    Plot the scatter diagram of the above data on graph paper and comment on the relationship between the two variables. 

(b)    Find the equation of the estimated least squares regression line of y on x and draw the line on your scatter diagram. 

 (c)   Calculate the correlation coefficient and state what this indicates about the relation between the two variables.

 (d)    The owner asks you to use the regression equation to forecast the daily sales if there were 20 hours of sunshine. Give two reasons why it would be inappropriate to do this.

(e)   Estimate the number of ice creams that would be sold if the number of hours of sunshine is 12. Comment on the reliability of the estimate.

Answer 1

Using desmos, I put the data points in as a table and then made a function that looks like this:

y₁ ~ mx₁ + b (to get the subscript, type "y" then underscore (shift dash) then the number "1")

This gives you the line of best fit (least squares regression line)

It looks like this:

It tells you the following statistics:

r² = 0.8485

r = 0.9211

and the following parameters:

m = 25.2773

b = 120.691

My comment might be that there is a medium-high positive linear correlation in the data.

From desmos, the equation that models the data would be y = 25.2773x + 120.691 again from desmos the correlation coefficient (r) is 0.9211 which means there is a good (I said medium-high before) positive linear correlation.

If there where 20 hours of sunlight, according to the modeling equation there would be:

y = 25.2773x + 120.691

y = 25.2773(20) + 120.691

y = 626.237

So there would be approximately 626 ice creams sold. I would say two reason why it would be inappropriate to do this are:

1) there isn't very much data

2) 20 is outside the reasonable data range (there can't be 20 hours of sunshine in a day)

If we try to project the number of ice creams sold for 12 hours we would get:

y = 25.2773x + 120.691

y = 25.2773(12) + 120.691

y = 424.0186

So there would be approximately 424 ice creams sold. I would say this could be a reasonable estimate.