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:

1. 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.
   
   

2. 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:

1. 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.
   

2. 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 r² 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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