Inferences on the Regression Line
Estimating the Mean Value of y for a Particular Value of x
Suppose that you own a pizza restaurant and are interesting in sending out
menus to local residents. You research what your 200 competitors have
done to find the relationship between number of mailings and amount of pizzas
bought per week. You find that the equation of the regression line
is
y = 100 + .2x.
You calculate s_{e} to be 4 and s_{a +
bx } to be 3. There are two things you are interested
in:

Next week you plan an advertising blitz of 1000 mailings. How
many pizzas do you expect to sell and what is a 95% confidence interval for
this estimate.

After next week you will be consistently sending out 300 mailings.
Over the next several years, what do you expect the average number
of pizzas sold will be?
Solutions:

We will use the main theorem that states that an unbiased estimate
for the value of y given a
fixed value of x is
a + bx The standard
deviation is
Hence we predict that we will sell about
100 + .2(1000) = 300 pizzas.
We
construct a confidence interval as
Hence we expect between 290 and 310 pizzas to be sold.

Now we are looking for the average y given a fixed value
of x. As before, we use the regression line for an unbiased estimator
for the mean. Hence we can expect to sell
100 + .2(300) = 160 pizzas.
for a confidence interval we use the standard deviation s_{a + bx
} to get the confidence interval:
160
1.96(3)
We can conclude with 95% confidence that we will average between 154 and
166 pizzas over the years.
Inferences on r
Recall that if there is no relationship between x and
y, then the correlation
= 0 and using the regression line is useless. If there is no relationship
then we call the variables independent. We can test to see if the
correlation is 0 with the following:
Ho: r = 0
Ha: r
0
We use the Greek letter r to indicate the population correlation. The
test statistic we use is
Notice that when r^{2} is close to
0, the test statistic is smaller.
Remark: This test is equivalent to the test for
b = 0.
Example
45 people were questioned to see if the
distance they lived from work was correlated to the distance they lived from
their parents. It was found that r = .8.
We calculate that
Since this is off the chart, we can conclude that there is a correlation
between distance from work and distance from parents house.
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