Question

Researchers asked each child in a sample of 411 school-age children if they were more or less likely to purchase a lottery ticket at a store if lottery tickets were visible on the counter. The percentage that said that they were more likely to purchase a ticket by grade level are as follows (R&J Child Development Consultants, Quebec, 2001):




Grade Percentage That Said They
Were More Likely to Purchase

6 32.7
8 46.1
10 75.0
12 83.6

a. Construct a scatterplot of y = percentage who said
they were more likely to purchase and x = grade. Does
there appear to be a linear relationship between x and y?
b. Find the equation of the least-squares line

Answer 1

Answer:

a) Figure attached

b) $y=9.08 x -22.37$

See calculations and explanation below.

Step-by-step explanation:

For this case we have the following data given:

Grade   Percentage That Said They  Were More Likely to Purchase

6                                                   32.7

8                                                   46.1

10                                                 75.0

12                                                 83.6

Part a

We denote:

y = percentage who said  they were more likely to purchase and x = grade

We can construct the scatterplot with Excel.

the result is on the figure attached

Part b

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i =36$

$\sum_{i=1}^n y_i =237.4$

$\sum_{i=1}^n x^2_i =344$

$\sum_{i=1}^n y^2_i =15808.46$

$\sum_{i=1}^n x_i y_i =2318.2$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=344-\frac{36^2}{4}=20$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=2318.2-\frac{36*237.4}{4}=181.6$

And the slope would be:

$m=\frac{181.6}{20}=9.08$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{36}{4}=9$

$\bar y= \frac{\sum y_i}{n}=\frac{237.4}{4}=59.35$

And we can find the intercept using this:

$b=\bar y -m \bar x=59.35-(9.08*9)=-22.37$

So the line would be given by:

$y=9.08 x -22.37$