**Television Ads:**

You own a chain of stores that sells television sets and you want

to know whether your advertising is increasing your sales. Let Y be the number of TVs you sell in a given month, and let X be the amount of money you spend on advertising in a given month in thousands of dollars.

You have data on advertising expenditures and sales for n=42 months and

have fit a simple linear regression of Y on X.

(d) Suppose your company makes a $100 profit per television sold BEFORE taking advertising costs into account. According to your best estimate, do the ads appear to be paying for themselves?

Our best estimate is that B1 = 10.2457. In other words, for every extra $1000 spent on advertising we sell an extra 10.2457 TVs. Since we make a profit of $100 per TV before advertising expenses this means every $1000 we spend on advertising results in an extra $1024.57 in TV sales. The sales exceed the costs of the ads–barely!–so our best guess is that the ads are

paying for themselves. From part (e) all we can say is that we are 99% sure that we have increased our sales between $883 and $1151 for each $1000 spent on ads.

The regression equation is: TV Sales = 48.4 + 10.2 Ad-Spending

Parameter Estimates

Predictor Coef Stdev t-ratio p

Constant 48.40 17.61 2.75 0.009

Ad Spending 10.2457 0.5224 19.61 0.000

RMSE = 38.54 R-sq = 90.6% R-sq(adj) = 90.3%

Analysis of Variance

SOURCE DF SS MS F p

Regression 1 571411 571411 384.70 0.000

Error 40 59413 1485

Total 41 630824

Summary Statistics For Number of Televisions Sold Per Month

N MEAN MEDIAN STDEV MIN MAX

TV Sales 42 373.5 363.0 124.0 146.0 609.0

(a) What percentage of variability in television sales is explained by advertising expenditures? Does the model do a good job in this respect?

The percentage of variability explained is given by R2 = 90.6% or R2 adj = 90.3%. The second number is more accurate since it takes the degrees of freedom into account. However, we accepted either one. They are actually very similar in this case–the

regression has explained just over 90% of the variability in television sales. Since the maximum

is 100% this seems like a pretty high amount. I would say the regression is doing a good job of

explaining the variability in sales.

(c) Is there a significant linear relationship between sales, Y, and advertising, X?

H0 : B1 = 0 There isn’t a significant linear relationship between advertising expenditures and television sales.

HA : B1 =/ 0 There is a significant linear relationship between advertising expenditures and televisions sales. How much you spend on advertising does help explain how many TVs you sell.

The corresponding p-value = .000. This is certainly smaller than = .005 so we reject

the null hypothesis. We conclude that there is a significant linear relationship between advertising expenditures and television sales. Knowing how much you spend on advertising does tell you something about what your sales will be like. In fact, the relationship is positive so spending more on ads is associated with higher sales, just as we would hope.

(b) Do advertising expenditures give good predictions for the number of television sales?

The average error is RMSE =38.54. Since Y is television sales, this means that when we try to predict the number of TVs sold each month we are off by about 38 sets. We sell around 373.5 TVs per month with a low one month of 146 and a high another

month of 609. So it looks like typically we make an error of around 10% (38.54/373.5) in our predictions.

**Linear Regression**

STAT 202

STAT 202

**Julieta Ruiz**

Liudmila Ruiz

Liudmila Ruiz

Let “X” is the average number of employees in a group insurance plan. Let “Y” be the average administrative cost as a percent of claims