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Chapter 3: Problem 2

Each month, the owner of Fay's Tanning Salon records in a data file the monthly total sales receipts and the amount spent that month on advertising. a. Identify the two variables. b. For each variable, indicate whether it is quantitative or categorical. c. Identify the response variable and the explanatory variable.

Short Answer

Expert verified
Variables: sales receipts, advertising amount; Both quantitative; Response: sales receipts; Explanatory: advertising amount.

Step by step solution

01

Identify the Variables

The exercise provides two main elements recorded: the monthly total sales receipts and the amount spent that month on advertising. The two variables involved are "monthly total sales receipts" and "amount spent on advertising."
02

Determine Variable Type for Sales Receipts

The "monthly total sales receipts" are amounts expressed in numbers (often currency), which can be measured or ordered quantitatively. Therefore, this variable is quantitative.
03

Determine Variable Type for Advertising Amount

Similarly, the "amount spent on advertising" is also a numeric value, indicating a measurable expenditure in currency. Hence, this variable is quantitative as well.
04

Identify the Response Variable

The response variable is typically the outcome we are interested in predicting or understanding. In this scenario, the interest is in the sales achieved, making "monthly total sales receipts" the response variable.
05

Identify the Explanatory Variable

The explanatory variable is used to explain or influence changes in the response variable. Here, the "amount spent on advertising" is intended to influence or predict sales and is thus the explanatory variable.

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Key Concepts

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

Quantitative Variable
In statistics, a quantitative variable refers to a variable that is numerical in nature. This means it takes values that can be measured and ordered. Quantitative variables can be further divided into continuous or discrete types. Continuous variables can take any numerical value within a range, such as height or temperature, while discrete variables are countable, like the number of students in a class.

In the context of Fay's Tanning Salon, both the "monthly total sales receipts" and the "amount spent on advertising" are quantitative variables.
  • "Monthly total sales receipts" are expressed in currency and represent a measurable outcome.
  • "Amount spent on advertising" is also a financial figure, indicating a specific expenditure.
Both variables are critical for analyzing the salon's financial performance. These quantitative measures allow for calculations, such as averages or sums, providing insights into business trends over time.
Explanatory Variable
The explanatory variable, often termed as an independent variable, is used to explain variations in another related variable known as the response variable. It is often the variable that a researcher manipulates or observes to assess its impact on the effect.

At Fay's Tanning Salon, the "amount spent on advertising" is designated as the explanatory variable. This is because:
  • It is assumed to have some influence over the business outcomes, specifically the monthly sales.
  • By analyzing changes in advertising spending, the salon can assess how it potentially impacts sales.
Understanding the role of an explanatory variable is crucial in determining cause-and-effect relationships within data sets.
Response Variable
A response variable, often known as a dependent variable, represents the primary outcome of interest in statistical studies. It is what the researcher aims to explain or predict based on variations in the explanatory variable.

In the exercise concerning Fay's Tanning Salon, the "monthly total sales receipts" serve as the response variable. This variable is critical, as it reflects the financial performance of the salon:
  • It is influenced by other factors, such as the amount spent on advertising.
  • Business decisions, like adjusting advertising budgets, are often made with the goal of impacting this response variable.
By focusing on the relationship between advertising and sales, the salon can aim to optimize its marketing strategies to drive better business outcomes.

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Most popular questions from this chapter

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.

Identify the values of the \(y\) -intercept \(a\) and the slope \(b\), and sketch the following regression lines, for values of \(x\) between 0 and 10 a. \(\hat{y}=7+0.5 x\) b. \(\hat{y}=7+x\) c. \(\hat{y}=7-x\) d. \(\hat{y}=7\)

In a survey conducted in March 2013 by the National Consortium for the Study of Terrorism and Responses to Terrorism, 1515 adults were asked about the effectiveness of the government in preventing terrorism and whether they believe that it could eventually prevent all major terrorist attacks. \(37.06 \%\) of the 510 adults who consider the government to be very effective believed that it can eventually prevent all major attacks, while this proportion was \(28.36 \%\) among those who consider the government somewhat, not too, or not at all effective in preventing terrorism. The other people surveyed considered that terrorists will always find a way. a. Identify the response variable, the explanatory variable and their categories. b. Construct a contingency table that shows the counts for the different combinations of categories. c. Use a contingency table to display the percentages for the categories of the response variables, separately for each category of the explanatory variable. d. Are the percentages reported in part c conditional? Explain. e. Sketch a graph that compares the responses for each category of the explanatory variable. fo Compute the difference and the ratio of proportions. Interpret. g. Give an example of how the results would show that there is no evidence of association between these variables.

In \(2014,\) the statistical summary of a weight loss survey was created and published on www.statcrunch.com. a. In this study, it seemed that the desired weight loss (in pounds) was a good predictor of the expected time (in weeks) to achieve the desired weight loss. Do you expect \(r^{2}\) to be large or small? Why? b. For this data, \(r=0.607\). Interpret \(r^{2}\). c. Show the algebraic relationship between the correlation of 0.607 and the slope of the regression equation \(b=0.437,\) using the fact that the standard deviations are 20.005 for pounds and 14.393 for weeks. (Hint: Recall that \(\left.=r \frac{s_{y}}{s_{x}} .\right)\)

According to data selected from GSS in \(2014,\) the correlation between \(y=\) email hours per week and \(x=\) ideal number of children is -0.0008 a. Would you call this association strong or weak? Explain. b. The correlation between email hours per week and Internet hours per week is \(0.33 .\) For this sample, which explanatory variable, ideal number of children or Internet hours per week, seems to have a stronger association with \(y ?\) Explain.
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