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Chapter 9: Problem 1

When drawing a scatter diagram, along which axis is the explanatory variable placed? Along which axis is the response variable placed?

Short Answer

Expert verified
Explanatory variable on x-axis; response variable on y-axis.

Step by step solution

01

Understanding Variables in a Scatter Diagram

In a scatter diagram, we typically deal with two types of variables: the explanatory variable and the response variable. The explanatory variable is the one that we suspect has an influence on or explains changes in another variable, often considered the 'independent variable.' The response variable is the outcome or 'dependent variable' that we are interested in observing or predicting changes in.
02

Determining the Axes for Variables

The explanatory variable, also known as the predictor or independent variable, is conventionally placed on the horizontal axis, also known as the x-axis, of the scatter diagram. This is because the x-axis is often used to present the variable that is being manipulated or controlled in an experimental study.
03

Placing the Response Variable

The response variable, also referred to as the dependent variable, is placed on the vertical axis, or y-axis, of the scatter diagram. This positioning allows us to see how changes in the explanatory variable affect the response variable, showing potential relationships or correlations between the two.

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Key Concepts

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

Explanatory Variable
An explanatory variable is like the guiding star in our scatter diagram. Imagine you have a garden and you want to understand how different amounts of water affect plant growth. The amount of water you give is the explanatory variable. This is because it is the variable you change to see what happens. You may hear this called the independent variable because it doesn't depend on other factors in the relationship you are studying, at least for the purpose of the scatter plot analysis.
  • Guides the experiment
  • Does not change based on the other variable
  • Sometimes labeled as the 'predictor' variable
In the context of a scatter diagram, the explanatory variable is always placed on the x-axis. This is the horizontal axis you see at the bottom of your graph. Putting it there allows us to clearly see how this variable possibly influences another, across different data points.
Response Variable
The response variable is where your curiosity lies. Continuing with the garden example, you would watch the plants grow, taking measurements to determine their height after different amounts of water. The height of the plants is the response variable. It literally responds to changes in the explanatory variable, which in this case is the amount of water.
  • Measures outcome
  • Depends on the explanatory variable
  • Also known as the 'dependent' variable
When you plot these on a scatter diagram, you put the response variable on the y-axis, the vertical axis on the side of your graph. This helps to graphically observe how different levels of the explanatory variable result in changes in the response, making it easy to detect patterns or trends.
Axes Placement in Statistics
Choosing the correct axes for your variables is crucial in any scatter diagram. The axes serve as guides, helping observers understand the relationship between the explanatory and response variables with clarity and precision. When thinking about axes: - **X-axis (Horizontal):** This is where the explanatory variable resides. It is the stage for our experiment, showing the range of values that predictor might take. - **Y-axis (Vertical):** This is where changes manifest, featuring the response variable. It allows us to track how the outcome shifts with different inputs from the x-axis. These placements are not just random; they align with conventional statistical methods, ensuring that the scatter plot communicates information in a standard format. This format makes it easier for anyone analyzing the data to immediately grasp the influence dynamics and possible correlations at play.

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Most popular questions from this chapter

The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let \(x\) be the number of different research programs, and let \(y\) be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs: $$ \begin{array}{l|rrrrrr} \hline x & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline y & 1.8 & 1.7 & 1.5 & 1.4 & 1.0 & 0.7 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=90, \Sigma y=8.1, \Sigma x^{2}=1420\), \(\Sigma y^{2}=11.83, \Sigma x y=113.8\), and \(r \approx-0.973 .\) (f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for \(y=\) mean number of patents per program?

When we take measurements of the same general type, a power law of the form \(y=\alpha x^{\beta}\) often gives an excellent fit to the data. A lot of research has been conducted as to why power laws work so well in business, economics, biology, ecology, medicine, engineering, social science, and so on. Let us just say that if you do not have a good straight-line fit to data pairs \((x, y)\), and the scatter plot does not rise dramatically (as in exponential growth), then a power law is often a good choice. College algebra can be used to show that power law models become linear when we apply logarithmic transformations to both variables. To see how this is done, please read on. Note: For power law models, we assume all \(x>0\) and all \(y>0\). Suppose we have data pairs \((x, y)\) and we want to find constants \(\alpha\) and \(\beta\) such that \(y=\alpha x^{\beta}\) is a good fit to the data. First, make the logarithmic transformations \(x^{\prime}=\log x\) and \(y^{\prime}=\log y .\) Next, use the \(\left(x^{\prime}, y^{\prime}\right)\) data pairs and a calculator with linear regression keys to obtain the least-squares equation \(y^{\prime}=a+b x^{\prime} .\) Note that the equation \(y^{\prime}=a+b x^{\prime}\) is the same as \(\log y=a+b(\log x)\). If we raise both sides of this equation to the power 10 and use some college algebra, we get \(y=10^{a}(x)^{b}\). In other words, for the power law model, we have \(\alpha \approx 10^{a}\) and \(\beta \approx b\). In the electronic design of a cell phone circuit, the buildup of electric current \((\mathrm{Amps})\) is an important function of time (microseconds). Let \(x=\) time in microseconds and let \(y=\) Amps built up in the circuit at time \(x .\) $$ \begin{array}{l|lllll} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y & 1.81 & 2.90 & 3.20 & 3.68 & 4.11 \\ \hline \end{array} $$ (a) Make the logarithmic transformations \(x^{\prime}=\log x\) and \(y^{\prime}=\log y .\) Then make a scatter plot of the \(\left(x^{\prime}, y^{\prime}\right)\) values. Does a linear equation seem to be a good fit to this plot? (b) Use the \(\left(x^{\prime}, y^{\prime}\right)\) data points and a calculator with regression keys to find the least-squares equation \(y^{\prime}=a+b x^{\prime} .\) What is the sample correlation coefficient? (c) Use the results of part (b) to find estimates for \(\alpha\) and \(\beta\) in the power law \(y=\alpha x^{\beta}\). Write the power law giving the relationship between time and Amp buildup. Note: The TI-84Plus/TI-83Plus/TI- \(n\) spire calculators fully support the power law model. Place the original \(x\) data in list L1 and the corresponding \(y\) data in list L2. Then press STAT, followed by CALC, and scroll down to option A: PwrReg. The output gives values for \(\alpha, \beta\), and the sample correlation coefficient \(r\).

How much should a healthy Shetland pony weigh? Let \(x\) be the age of the pony (in months), and let \(y\) be the average weight of the pony (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used in most veterinary colleges). $$ \begin{array}{r|rrrrr} \hline x & 3 & 6 & 12 & 18 & 24 \\ \hline y & 60 & 95 & 140 & 170 & 185 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=63, \quad \Sigma x^{2}=1089, \quad \Sigma y=650\) \(\Sigma y^{2}=95,350\), and \(\Sigma x y=9930 .\) Compute \(r .\) As \(x\) increases from 3 to 24 months, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

All Greens is a franchise store that sells house plants and lawn and garden supplies. Although All Greens is a franchise, each store is owned and managed by private individuals. Some friends have asked you to go into business with them to open a new All Greens store in the suburbs of San Diego. The national franchise headquarters sent you the following information at your request. These data are about 27 All Greens stores in California. Each of the 27 stores has been doing very well, and you would like to use the information to help set up your own new store. The variables for which we have data are \(x_{1}=\) annual net sales, in thousands of dollars \(x_{2}=\) number of square feet of floor display in store, in thousands of square feet \(x_{3}=\) value of store inventory, in thousands of dollars \(x_{4}=\) amount spent on local advertising, in thousands of dollars \(x_{5}=\) size of sales district, in thousands of families \(x_{6}=\) number of competing or similar stores in sales district A sales district was defined to be the region within a 5 -mile radius of an All Greens store. $$ \begin{array}{rlrrrr|rrrrrr} \hline x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} \\ \hline 231 & 3 & 294 & 8.2 & 8.2 & 11 & 65 & 1.2 & 168 & 4.7 & 3.3 & 11 \\ 156 & 2.2 & 232 & 6.9 & 4.1 & 12 & 98 & 1.6 & 151 & 4.6 & 2.7 & 10 \\ 10 & 0.5 & 149 & 3 & 4.3 & 15 & 398 & 4.3 & 342 & 5.5 & 16.0 & 4 \\ 519 & 5.5 & 600 & 12 & 16.1 & 1 & 161 & 2.6 & 196 & 7.2 & 6.3 & 13 \\ 437 & 4.4 & 567 & 10.6 & 14.1 & 5 & 397 & 3.8 & 453 & 10.4 & 13.9 & 7 \\ 487 & 4.8 & 571 & 11.8 & 12.7 & 4 & 497 & 5.3 & 518 & 11.5 & 16.3 & 1 \\ 299 & 3.1 & 512 & 8.1 & 10.1 & 10 & 528 & 5.6 & 615 & 12.3 & 16.0 & 0 \\ 195 & 2.5 & 347 & 7.7 & 8.4 & 12 & 99 & 0.8 & 278 & 2.8 & 6.5 & 14 \\ 20 & 1.2 & 212 & 3.3 & 2.1 & 15 & 0.5 & 1.1 & 142 & 3.1 & 1.6 & 12 \\ 68 & 0.6 & 102 & 4.9 & 4.7 & 8 & 347 & 3.6 & 461 & 9.6 & 11.3 & 6 \\ 570 & 5.4 & 788 & 17.4 & 12.3 & 1 & 341 & 3.5 & 382 & 9.8 & 11.5 & 5 \\ 428 & 4.2 & 577 & 10.5 & 14.0 & 7 & 507 & 5.1 & 590 & 12.0 & 15.7 & 0 \\ 464 & 4.7 & 535 & 11.3 & 15.0 & 3 & 400 & 8.6 & 517 & 7.0 & 12.0 & 8 \\ 15 & 0.6 & 163 & 2.5 & 2.5 & 14 & & & & & & \\ \hline \end{array} $$ (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section \(3.2\) ) for each variable. Relative to its mean, which variable has the largest spread of data values? Which variable has the least spread of data values relative to its mean? (b) For each pair of variables, generate the sample correlation coefficient \(r .\) For all pairs involving \(x_{1}\), compute the corresponding coefficient of determination \(r^{2}\). Which variable has the greatest influence on annual net sales? Which variable has the least influence on annual net sales? (c) Perform a regression analysis with \(x_{1}\) as the response variable. Use \(x_{2}, x_{3}\), \(x_{4}, x_{5}\), and \(x_{6}\) as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in \(x_{1}\) can be explained by the corresponding variations in \(x_{2}, x_{3}, x_{4}, x_{5}\), and \(x_{6}\) taken together? (d) Write out the regression equation. If two new competing stores moved into the sales district but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain your answer. If you increased the local advertising by a thousand dollars but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain. (e) Test each coefficient to determine if it is or is not zero. Use level of significance \(5 \%\). (f) Suppose you and your business associates rent a store, get a bank loan to start up your business, and do a little research on the size of your sales district and the number of competing stores in the district. If \(x_{2}=2.8\), \(x_{3}=250, x_{4}=3.1, x_{5}=7.3\), and \(x_{6}=2\), use a computer to forecast \(x_{1}=\) annual net sales and find an \(80 \%\) confidence interval for your forecast (if your software produces prediction intervals). (g) Construct a new regression model with \(x_{4}\) as the response variable and \(x_{1}\), \(x_{2}, x_{3}, x_{5}\), and \(x_{6}\) as explanatory variables. Suppose an All Greens store in Sonoma, California, wants to estimate a range of advertising costs appropriate to its store. If it spends too little on advertising, it will not reach enough customers. However, it does not want to overspend on advertising for this type and size of store. At this store, \(x_{1}=163, x_{2}=2.4, x_{3}=188\), \(x_{5}=6.6\), and \(x_{6}=10\). Use these data to predict \(x_{4}\) (advertising costs) and find an \(80 \%\) confidence interval for your prediction. At the \(80 \%\) confidence level, what range of advertising costs do you think is appropriate for this store?

In the least squares line \(\hat{y}=5+3 x\), what is the marginal change in \(\hat{y}\) for each unit change in \(x\) ?
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