Unit 5 · 10–20% of the exam Beta

Unit 5 practice test: Regression Analysis

12 exam-style questions from the unit, free, with an explanation for every option. Most have four answer choices, as on the real exam, and a few ask you to type a value, an interval or the setup; sets share one data set, plot or table.

Covers: Graphical Representations Between Two Quantitative Variables; Correlation; Linear Regression Models; Residuals; Least-Squares Regression.

Graph

Four scatterplots

Each plot shows 20 points of invented data, with its correlation computed from the data printed above it.

Four small scatterplots with their correlations computed from the data: strong positive, r = 0.99; moderate negative, r = -0.79; weak positive, r = 0.29; and a strong curved arch with r = 0.04. — LevlPrep original diagram.

1. Which plot shows a strong relationship even though its correlation is close to 0?

The strong positive plot, $r = 0.99$
The weak positive plot, $r = 0.29$
The moderate negative plot, $r = -0.79$
The curved plot, $r = 0.04$

Show the answer

Correlation near 0 means no linear association; there may still be a strong curved one.

The strong positive plot, $r = 0.99$ : Its $r$ is close to 1, not 0.
The weak positive plot, $r = 0.29$ : Those points are widely scattered; the relationship really is weak.
The moderate negative plot, $r = -0.79$ : Its $r$ is far from 0.
Correct: The curved plot, $r = 0.04$ : Right: the points follow an arch very closely, but $r$ measures only linear association, and an arch is not a line.

2. Which plot has the strongest linear association?

The plot with $r = 0.29$ , because its points spread widest
The plot with $r = 0.99$
The plot with $r = 0.04$ , because its pattern is clear
The plot with $r = -0.79$ , because negative is stronger

Show the answer

Linear strength is the size of $r$ , $|r|$ , whatever its sign.

The plot with $r = 0.29$ , because its points spread widest: Wide spread means a weak association.
Correct: The plot with $r = 0.99$ : Right: its $r$ is closest to 1 or $-1$ in size, and its points hug a line.
The plot with $r = 0.04$ , because its pattern is clear: The curve is clear, but it is not linear, and $r$ measures linear strength.
The plot with $r = -0.79$ , because negative is stronger: The sign gives direction only. Compare sizes: 0.99 is larger than 0.79.

3. What does the plot with $r = -0.79$ show?

A weak association, because $r$ is negative
That $y$ decreases by 0.79 for each unit of $x$
That 79% of the points lie on a line
A moderate linear trend: $y$ tends to fall as $x$ rises

Show the answer

Sign: direction. Size: strength of the linear pattern.

A weak association, because $r$ is negative: A negative $r$ is not weaker; $-0.79$ is as strong as 0.79.
That $y$ decreases by 0.79 for each unit of $x$ : That would be a slope; $r$ has no units.
That 79% of the points lie on a line: $r$ is not a percent of points.
Correct: A moderate linear trend: $y$ tends to fall as $x$ rises: Right: the negative sign gives the direction, and a size of 0.79 is a moderate linear association, close to strong.

4. A student says, "The curved plot has $r = 0.04$ , so $x$ and $y$ are unrelated." What should you tell the student?

The student is right: a correlation near 0 means no relationship of any kind
The student should square $r$ to see the true strength
The correlation was probably computed wrong
Look at the plot first: $r$ misses curved relationships

Show the answer

Correlation describes linear relationships only.

The student is right: a correlation near 0 means no relationship of any kind: A strong curve can have $r$ near 0, as this plot shows.
The student should square $r$ to see the true strength: Squaring a small number makes it smaller and still ignores the curve.
The correlation was probably computed wrong: It is correct; the arch is symmetric, so the upward and downward halves cancel in $r$ .
Correct: Look at the plot first: $r$ misses curved relationships: Right: always plot the data; a near-zero $r$ rules out only a linear pattern.

Graph

Practice and free throws, with one unusual player

A coach records practice hours per week and free throw percentage for 11 players. One player practices 16 hours a week and makes 58% of free throws. (Invented data.)

5. How is the player at 16 hours best described?

A typical point, since it lies inside the range of $y$ values
A point with a large residual that the line passes far from
An outlier with no effect on the line, since its percentage is ordinary
High leverage and influential: extreme $x$ , and it pulls the line

Show the answer

High leverage: an extreme $x$ value. Influential: removing it changes the line a lot.

A typical point, since it lies inside the range of $y$ values: Its $x$ value, 16, is twice the next largest, which makes it unusual.
A point with a large residual that the line passes far from: The line is pulled close to it, so its residual is not very large; that is what influence looks like.
An outlier with no effect on the line, since its percentage is ordinary: Its 58% is ordinary on its own, but at 16 hours it is far from the others' $x$ values, so it moves the line a lot.
Correct: High leverage and influential: extreme $x$ , and it pulls the line: Right: far from $\bar{x}$ gives high leverage, and removing it changes the slope a lot.

6. With all 11 players the least-squares slope is about 0.49. What happens to the slope if the 16-hour player is removed?

It becomes negative
It decreases to about 0.24
It stays about 0.49, since one point out of 11 is small
It increases to about 2.92

Show the answer

Removing an influential point changes the slope substantially.

It becomes negative: The other ten points rise steeply; the slope without the point is positive and large.
It decreases to about 0.24: The unusual point is below where the trend of the others would put it, so removing it makes the line steeper, not flatter.
It stays about 0.49, since one point out of 11 is small: A point with high leverage can change the slope a great deal, as here.
Correct: It increases to about 2.92: Right: technology gives $b = 2.92$ for the other ten players. The one point had flattened the line.

7. With all 11 players, $r = 0.32$ . Without the 16-hour player, $r = 0.97$ . What does this show?

Practice has little to do with free throws for these players
The 16-hour player should be deleted from any report
One point hides a strong linear link among the other ten
Correlation is resistant to unusual points

Show the answer

Unusual points can change $r$ and the slope; analyze with and without them.

Practice has little to do with free throws for these players: Among the ten typical players the association is strong; the weak $r$ is caused by one point.
The 16-hour player should be deleted from any report: Report the point and the analysis with and without it; do not delete data without a reason.
Correct: One point hides a strong linear link among the other ten: Right: correlation is not resistant; one high-leverage point weakened it from 0.97 to 0.32.
Correlation is resistant to unusual points: The change from 0.97 to 0.32 shows the opposite.

8. Which pair of variables could be shown on a scatterplot?

Height and arm span of 25 students, both in centimeters
Grade level and favorite subject of 25 students
Heights of 25 students in one class and 30 students in another
Favorite sport and eye color of 25 students

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A scatterplot needs two quantitative variables on the same individuals.

Correct: Height and arm span of 25 students, both in centimeters: Right: two quantitative variables measured on the same individuals.
Grade level and favorite subject of 25 students: Favorite subject is categorical, so a scatterplot cannot show it.
Heights of 25 students in one class and 30 students in another: A scatterplot pairs two variables on the same individuals. These are one variable for two groups.
Favorite sport and eye color of 25 students: Both are categorical. A two-way table or segmented bar chart fits them.

9. For children aged 5 to 12, a line predicts height (cm) from age (years): $\hat{y} = 76 + 6.2x$ . Why should the intercept not be interpreted as a newborn's height?

Age 0 is outside the data, and growth is not linear from birth
Intercepts are not meaningful in any regression line
76 cm is the average height of the children in the study
The slope is positive, so the intercept should be ignored

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Interpret the intercept only when $x = 0$ makes sense in context.

Correct: Age 0 is outside the data, and growth is not linear from birth: Right: interpreting the intercept here would be extrapolation; real newborns are about 50 cm.
Intercepts are not meaningful in any regression line: An intercept is meaningful when $x = 0$ is inside the data, as for new phones.
76 cm is the average height of the children in the study: The intercept is the prediction at age 0, not an average.
The slope is positive, so the intercept should be ignored: The sign of the slope does not decide whether the intercept makes sense.

10. A residual plot shows residuals spread out more and more as $x$ increases, like a fan. What does this mean?

The relationship is curved
The correlation is exactly 0
Predictions are less precise for large $x$ , since the scatter grows
The line fits perfectly for large $x$

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Fan shape: spread of the residuals changes with $x$ .

The relationship is curved: A curve shows up as a bend in the residuals. A fan shows growing spread.
The correlation is exactly 0: A fan shape says nothing about $r$ being 0.
Correct: Predictions are less precise for large $x$ , since the scatter grows: Right: the line's typical error changes with $x$ , so predictions for large $x$ are less reliable.
The line fits perfectly for large $x$ : The residuals are largest there, so the fit is worst there.

11. Across cities, the number of fire trucks sent to a fire and the dollar damage have a correlation of 0.7. What explains this best?

70% of the damage is caused by the trucks
Bigger fires bring more trucks and more damage
The correlation is probably a mistake in the data
The fire trucks themselves cause most of the damage

Show the answer

A strong correlation can come from a lurking variable.

70% of the damage is caused by the trucks: $r$ is not a percent of anything.
Correct: Bigger fires bring more trucks and more damage: Right: fire size is a lurking variable linked to both. Correlation does not mean the trucks cause damage.
The correlation is probably a mistake in the data: The correlation is real; its explanation is a lurking variable.
The fire trucks themselves cause most of the damage: Correlation is not causation; a third variable explains both.

12. A scatterplot has correlation $r = -0.7$ . What is true of the slope of its regression line?

It is negative
It is $-0.7$
It is positive
It is 0

Show the answer

Slope and correlation share their sign.

Correct: It is negative: Right: the slope and the correlation always have the same sign.
It is $-0.7$ : The slope has units and depends on the spreads of $x$ and $y$ ; it equals $r$ only in special cases.
It is positive: A negative correlation means $y$ tends to fall as $x$ rises, so the line falls.
It is 0: A slope of 0 would mean no linear trend, but $r = -0.7$ shows one.

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Practice has every question in the unit, with feedback after each one; the notes for every topic are free. Other units: Unit 1 · Unit 2 · Unit 3 · Unit 4 · Unit 5.