A scatter plot of price against living area shows points rising steadily with little vertical spread. What does this indicate?
Correct Answer
A) A strong positive relationship between the two
Why this is correct: A strong positive relationship between the two. The governing concept is that a scatter plot shows the relationship between two variables. Points rising steadily with little vertical spread indicate that as living area increases, price increases predictably, showing a strong positive linear correlation. Why the other choices are wrong: That living area does not influence sale price is wrong because the rising pattern shows clear influence. That the sample contains a recording error is wrong; tight clustering suggests data consistency, not error. A negative relationship between the variables is wrong; a negative relationship would show a downward trend. Exam tip: A tight, upward-sloping cluster on a scatter plot visually confirms a strong positive relationship before any statistical calculation.
Why This Is the Correct Answer
Rising points indicate a positive relationship and tight vertical spread indicates a strong one, so the plot shows a strong positive relationship between living area and price. That pattern is what would justify using square footage as a primary variable in a regression or as a unit of comparison in the grid. It also implies that price per square foot will be relatively stable across the sample, which supports its use as a comparison metric. Correlation of this kind is expected within a well-defined market segment.
Why the Other Options Are Wrong
Option B: That living area does not influence sale price
No influence would appear as a horizontal band or a formless cloud, where knowing the living area tells you nothing about the price. The stem describes points that rise steadily, which is the visual signature of exactly the opposite. Reading a clear pattern as no relationship inverts the plot's meaning.
Option C: That the sample contains a recording error
Tight clustering around a trend is a sign of consistent, well-behaved data, not of error. Recording errors typically appear as isolated outliers far from the pattern, such as a point at ten times the price or a tenth of the area. The option reverses the diagnostic significance of low scatter.
Option D: A negative relationship between the variables
A negative relationship would slope downward, meaning larger homes sold for less, which would be a striking finding worth investigating for a mix problem or a data error. The stem explicitly says the points rise. Direction is the easiest thing to read off a scatter plot and the first thing to check.
Slope for Direction, Spread for Strength
Two questions, two features. Which way does the cloud lean? That is the sign. How thick is the cloud? Thin means strong, fat means weak. Answer both and you have described the relationship.
How to use: Read the stem for words about direction, rising or falling, and words about spread, tight or scattered. Combine them into one phrase and match it to the option.
Exam Tip
Correlation is not causation, and a strong plot does not prove living area causes price. It shows association, which is enough to support a unit of comparison but not a causal claim.
Common Mistakes to Avoid
- -Computing a correlation coefficient without plotting the data first
- -Reading correlation as causation
- -Fitting a straight line to a relationship that curves across a wide size range
Concept Deep Dive
Analysis
A scatter plot is the first diagnostic an analyst runs before computing anything, because the eye catches patterns that summary statistics can hide. Two features of the cloud carry the information. Direction is the slope of the pattern: rising left to right means the two variables move together, which is a positive relationship, while falling means a negative one. Strength is the tightness of the points around that pattern: little vertical spread means the relationship explains most of the variation and predictions from it will be reliable, while a wide cloud means other factors dominate. Points rising steadily with little vertical spread therefore show a strong positive relationship, which is exactly what appraisers expect between gross living area and price within a homogeneous market segment. Plotting first also reveals things a correlation coefficient would miss: curvature, outliers, and clusters that signal two submarkets pooled together.
Background Knowledge
You need to read direction and strength from a scatter plot, and to know that correlation measures association rather than causation. You should also know that a scatter plot is used to check linearity, spot outliers, and detect clusters before fitting any model, and that price per square foot tends to decline as size increases across a wide size range.
Real-World Application
An appraiser plots price against living area for thirty-two sales in one subdivision, sees a tight upward band, and uses square footage as the primary size variable in her adjustment analysis. Two outliers well above the line turn out to have finished basements not counted in living area, which she handles as a separate adjustment.
More Statistics Questions
A set of comparable sales has a mean of $250,000 and a standard deviation of $20,000. What is the coefficient of variation?
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