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Chapter 3: Problem 35

Advertising and sales Each month, the owner of Fay's Tanning Salon records in a data file \(y=\) monthly total sales receipts and \(x=\) amount spent that month on advertising, both in thousands of dollars. For the first three months of operation, the observations are as shown in the table. \begin{tabular}{cc} \hline Advertising & Sales \\ \hline 0 & 4 \\ 1 & 6 \\ 2 & 8 \\ \hline \end{tabular} a. Sketch a scatterplot. b. From inspection of the scatterplot, state the correlation and the regression line. (Note: You should be able to figure them out without using software or formulas.) c. Find the mean and standard deviation for each variable. d. Using part c, find the regression line, using the formulas for the slope and the \(y\) -intercept. Interpret the \(y\) -intercept and the slope.

Short Answer

Expert verified
The regression line is \( y = 4 + 2x \), with a perfect positive correlation.

Step by step solution

01

Sketch the Scatterplot

Draw a graph with the horizontal axis labeled as 'Advertising (in thousands)' and the vertical axis labeled as 'Sales (in thousands)'. Plot the three data points from the table: (0, 4), (1, 6), and (2, 8). The points should appear to form a straight line, indicating a linear relationship.
02

Analyze the Scatterplot

From the scatterplot, we can visually observe that the relationship between advertising and sales forms a perfectly linear, upward-sloping line. This suggests that the correlation is positive and perfect, meaning the correlation coefficient is 1. The regression line can be observed directly as going through these points.
03

Calculate Means

The mean of the advertising values is \( \bar{x} = \frac{0 + 1 + 2}{3} = 1 \). The mean of the sales values is \( \bar{y} = \frac{4 + 6 + 8}{3} = 6 \).
04

Calculate Standard Deviations

For the advertising variable (\(x\)), the standard deviation is calculated as:\[ s_x = \sqrt{\frac{(0-1)^2 + (1-1)^2 + (2-1)^2}{2}} = \sqrt{1} = 1 \]For the sales variable (\(y\)), the standard deviation is:\[ s_y = \sqrt{\frac{(4-6)^2 + (6-6)^2 + (8-6)^2}{2}} = \sqrt{4} = 2 \]
05

Find the Regression Equation

Using the formula for the slope \(b\) of a regression line \(y = a + bx\), where \(b = \frac{s_y}{s_x} \cdot r\):Since the correlation \(r = 1\), slope \(b = \frac{2}{1} \cdot 1 = 2\).The y-intercept \(a\) is calculated using the point \((\bar{x}, \bar{y})\):\[ a = \bar{y} - b\bar{x} = 6 - 2 \times 1 = 4 \]Thus, the regression line is \( y = 4 + 2x \).
06

Interpretation of Slope and Intercept

The slope of 2 indicates that for every additional $1,000 spent on advertising, sales increase by $2,000. The y-intercept of 4 suggests that if nothing is spent on advertising, the expected sales would be $4,000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot Analysis
A scatterplot is a graph that shows the relationship between two variables. For Fay's Tanning Salon, our scatterplot helps us visualize the connection between advertising costs and sales receipts. To create it, we plot the advertising spending on the horizontal axis and the corresponding sales on the vertical axis. In this case, the data points are (0, 4), (1, 6), and (2, 8).

When we place these points on the graph, they form a straight line. This straight line indicates a linear relationship, meaning as the advertising spending increases, sales also increase consistently. This kind of perfect alignment tells us that the correlation is strong and positive. In fact, it is the strongest linear relationship possible, reflected by a correlation coefficient of 1. Such visual analysis through a scatterplot is essential in identifying relationships between variables without complex calculations.
Mean and Standard Deviation Calculation
The mean, also known as the average, helps us find a central value in a data set. For the tanning salon's advertising spending, the mean is calculated by adding all advertising spending values and dividing by the number of months. Mathematically, \( \bar{x} = \frac{0 + 1 + 2}{3} = 1 \). Similarly, the mean sales is \( \bar{y} = \frac{4 + 6 + 8}{3} = 6 \). These means indicate the average monthly spending on advertising and the resulting sales amount.

Next is the standard deviation, which describes how much the values in a data set differ from the mean. For advertising spending, the standard deviation is calculated as:\[ s_x = \sqrt{\frac{(0-1)^2 + (1-1)^2 + (2-1)^2}{2}} = 1 \]. For sales, it is \[ s_y = \sqrt{\frac{(4-6)^2 + (6-6)^2 + (8-6)^2}{2}} = 2 \]. This shows us that sales have more variability compared to advertising spending. The calculations for mean and standard deviation are crucial in understanding the data distribution, giving us insights into the average values and the spread in the data.
Linear Relationship Interpretation
Interpreting a linear relationship involves understanding how two variables interact in a straight-line fashion. With Fay's Tanning Salon, we use the regression equation, \( y = a + bx \), to express this relationship mathematically. The formula uses the slope \(b\) and the y-intercept \(a\). Here, \(b = 2\), derived from the ratio \[ \frac{s_y}{s_x} \cdot r \] and the fact that the correlation \(r\) is 1, showcasing a perfect relationship.

The slope of 2 implies that for every additional \(1,000 spent on advertising, sales are likely to increase by \)2,000. The y-intercept of 4 suggests that if no money is spent on advertising, sales would be expected to stand at $4,000. This intercept represents the baseline level of sales independent of advertising. Understanding these components of the regression equation is fundamental to predicting how changes in one variable affect another, which is valuable for business decisions and strategy planning.

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Most popular questions from this chapter

How much do seat belts help? In \(2013,\) data was collected from the U.S. Department of Transportation and the Insurance Institute for Highway Safety. According to the collected data, the number of deaths per 100,000 individuals in the U.S would decrease by 24.45 for every 1 percentage point gain in seat belt usage. Let \(\hat{y}=\) predicted number of deaths per 100,000 individuals in 2013 and \(x=\) seat belt use rate in a given state. a. Report the slope \(b\) for the equation \(\hat{y}=a+b x\). b. If the \(y\) intercept equals \(32.42,\) then predict the number of deaths per 100,000 people in a state if (i) no one wears seat belts, (ii) \(74 \%\) of people wear seat belts (the value for Montana), (iii) \(100 \%\) of people wear seat belts.

The figure shows recent data on \(x=\) the number of televisions per 100 people and \(y=\) the birth rate (number of births per 1000 people) for six African and Asian nations. The regression line, \(\hat{y}=29.8-0.024 x,\) applies to the data for these six countries. For illustration, another point is added at \((81,15.2),\) which is the observation for the United States. The regression line for all seven points is \(\hat{y}=31.2-0.195 x\). The figure shows this line and the one without the U.S. observation. a. Does the U.S. observation appear to be (i) an outlier on \(x,\) (ii) an outlier on \(y,\) or (iii) a regression outlier relative to the regression line for the other six observations? b. State the two conditions under which a single point can have a dramatic effect on the slope and show that they apply here. c. This one point also drastically affects the correlation, which is \(r=-0.051\) without the United States but \(r=-0.935\) with the United States. Explain why you would conclude that the association between birth rate and number of televisions is (i) very weak without the U.S. point and (ii) very strong with the U.S. point. d. Explain why the U.S. residual for the line fitted using that point is very small. This shows that a point can be influential even if its residual is not large.

Rating restaurants Zagat restaurant guides publish ratings of restaurants for many large cities around the world (see www.zagat.com). The review for each restaurant gives a verbal summary as well as a 0 - to 30 -point rating of the quality of food, d茅cor, service, and the cost of a dinner with one drink and tip. For 31 French restaurants in Boston in \(2014,\) the food quality ratings had a mean of 24.55 and standard deviation of 2.08 points. The cost of a dinner (in U.S. dollars) had a mean of \(\$ 50.35\) and standard deviation of \(\$ 14.92 .\) The equation that predicts the cost of a dinner using the rating for the quality of food is \(\hat{y}=-70+4.9 x\). The correlation between these two variables is 0.68 . (Data available in the Zagat_Boston file.) a. Predict the cost of a dinner in a restaurant that gets the (i) lowest observed food quality rating of 21 , (ii) highest observed food quality rating of 28 . b. Interpret the slope in context. c. Interpret the correlation. d. Show how the slope can be obtained from the correlation and other information given.

Internet and email use According to data selected from GSS in \(2014,\) the correlation between \(y=\) email hours per week and \(x=\) Internet hours per week is \(0.33 .\) The regression equation is predicted email hours \(=3.54+\) 0.25 Internet hours a. Based on the correlation value, the slope had to be positive. Why? b. Your friend says she spends 60 hours on the Internet and 10 hours on email in a week. Find her predicted email use based on the regression equation. c. Find her residual. Interpret.

Wage bill of Premier League Clubs Data of the Premier League Clubs' wage bills was obtained from www.tsmplug .com. For the response variable \(y=\) wage bill in millions of pounds in 2014 and the explanatory variable \(x=\) wage bill in millions of pounds in \(2013, \hat{y}=-1.537+1.056 x\). a. How much do you predict the value of a club's wage bill to be in 2014 if in 2013 the club had a wage bill of (i) \(拢 100\) million, (ii) \(拢 200\) million? b. Using the results in part a, explain how to interpret the slope. c. Is the correlation between these variables positive or negative? Why? d. A Premier League club had a wage bill of \(拢 100\) million in 2013 and \(拢 105\) million in \(2014 .\) Find the residual and interpret it.
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