Scatter Plots Explained

Make a scatterplot of the data. Use the year on the horizontal scale and the number of ounces on the vertical scale.
Available Drink Sizes at a Convenience Store Year, Sizes Available (cm)
1973: 12, 20
1976: 12, 16, 20
1978: 12, 16, 20, 32
1983: 12, 16, 20, 32, 44
1988: 12, 16, 20, 32, 44, 64
2003: 12, 20, 32, 44, 64
2005: 20, 32, 44, 64

How can two variables be displayed in a scattergram?

Suppose x1 and x2 are predictor variables for a response variable y.
a. The distribution of all possible values of the response variable corresponding to particular values of the two predictor variables is called a distribution of the response variable.
b. State the four assumptions for multiple linear regression inferences

Describe about the three positive relationships of Scatterplots?

Make a scatterplot for the data.
Length (mi) and Water Flow ${\displaystyle (1,000f\frac{t^3}{s})}$ of Rivers
Length: 2540, 1980, 1460, 1420, 1290, 1040, 886, 774, 724, 659
Flow: 76, 225, 41, 58, 56, 57, 68, 67, 67, 41

Predicting Land Value Both figures concern the assessed value of land (with homes on the land), and both use the same data set. (a). Which do you think has a stronger relationship with value of the land-the number of acres of land or the number of rooms n the homes? Why? b. Il you were trying to predict the value of a parcel of land in e arca (on which there is a home), would you be able to te a better prediction by knowing the acreage or the num- ber of rooms in the house? Explain. (Source: Minitab File, Student 12. "Assess.")

The following data on = soil depth (in centimeters) and y = percentage of montmorillonite in the soil were taken from a scatterplot in the paper "Ancient Maya Drained Field Agriculture: Its Possible Application Today in the New River Floodplain, Belize, C.A." (Agricultural Ecosystems and Environment [1984]: 67-84):
a. Draw a scatterplot of y versus x.
b. The equation of the least-squares line is 0.45x. Draw this line on your scatterplot. Do there appear to be any large residuals?
c. Compute the residuals, and construct a residual plot. Are there any unusual features in the plot?
$\begin{array}{|cccccccc|}\hline x& 40& 50& 60& 70& 80& 90& 100\\ y& 58& 34& 32& 30& 28& 27& 22\\ \hline\end{array}$
${\displaystyle [\hat{y}=64.50]}$.

Find the linear regression line for a scatterplot formed by the points ${\displaystyle (10,261),(21,252),(42,209),(33,163),}$ and ${\displaystyle (52,98).}$ Round to the nearest tenth.

Make a scatterplot for each set of data.
$\begin{array}{|ccccccccccc|}\hline \text{Hits:}& 7& 8& 4& 11& 8& 2& 5& 9& 1& 4\\ \text{Runs:}& 3& 2& 2& 7& 4& 2& 1& 3& 0& 1\\ \hline\end{array}$

Name only one data shape which is different from symmetric and right/left skewed
Name four different graphs that show a summary of the data.

The accompanying data on y = normalized energy \(\displaystyle{\left[{\left(\frac{{J}}{{m}^{{2}}}\right)}\right]}\) and x = intraocular pressure (mmHg) appeared in a scatterplot in the article “Evaluating the Risk of Eye Injuries: Intraocular Pressure During High Speed Projectile Impacts” (Current Eye Research, 2012: 43-49), an estimated regression function was superimposed on the plot.
\(\begin{array}{|c|c|}\hline x & 2761 & 19764 & 25713 & 3980 & 12782 & 19008 & 19028 & 14397 & 9606 & 3905 & 25731 \\ \hline y & 1553 & 14999 & 32813 & 1667 & 8741 & 16526 & 26770 & 16526 & 9868 & 6640 & 1220 & 30730 \\ \hline \end{array}\)
Here is Minitab output from fitting the simple linear regression model. Does the model appear to specify a useful relationship between the two variables?
\(\begin{array}{|c|c|}\hline \text{Predictor Coef SE Coef T P Constant} & -5090 & 2257 & -2.26 & 0.048 \\ \hline \text{Pressure} & 1.2912 & 0.1347 & 9.59 & 0.000 \\ \hline \end{array}\)
\([S=3679.36, R-Sq = 90.2\%, R-Sq(adj)=89.2\% ]\).

Make a scatterplot of the data and graph the function ${\displaystyle f\left(x\right)=-8x^2+95x+745.}$ Make a residual plot and describe how well the function fits the data. $\begin{array}{|cccccc|}\hline \text{Price Increase}& 0& 1& 2& 3& 4\\ \text{Sales}& 730& 850& 930& 951& 1010\\ \hline\end{array}$

State the slope and y-intercept of the equation. $5x+4y=20$

Scatterplots Which of the scatterplots below show
a) little or no association?
b) a negative association?
c) a linear association?
d) a moderately strong association?
e) a very strong association?

Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

In this exercise, you will use the correlation and regression applet to create scatter plots with 10 points that have a correlation close to 0.7. The lesson here is that many models may have the same correlation. Always compile your data before trusting correlations. (a) Stop after adding the first two points. What is the value of correlation?
(Enter your answer, rounded to four decimal places).
r=?
Why does correlation matter? Two is the minimum number of data points required to calculate the correlation. This value is the default correlation.
Because two points define a line, correlation always matters.
The mean of these two values always has this value.

Horace keeps track of the amount of time he studies and the score he gets on his quizzes. The data are shown in the table below. Which of the scatter plots below accurately records the data?

Regarding a scatterplot matrix:
a. Identify two of its uses.
b. What property should the scatterplots of the response versus each predictor variable have in order to proceed to obtain the regression plane for the data?

A researcher investigating the association between two variables collected some data and was surprised when he calculated the correlation.
He had expected to find a fairly strong association, yet the correlation was near 0.
Discouraged, he didnt

The scatterplots below display three bivariate data sets. The correlation coefficients for these data sets are 0.03, 0.68, and 0.89. Which scatter plot corresponds to the data set with r = .03?