Chapter 4: Linear Regression

What is a scatterplot?

A graph of paired numerical data shown as points.

What variable usually goes on the x-axis in a scatterplot?

The explanatory / independent variable.

What variable usually goes on the y-axis in a scatterplot?

The response / dependent variable.

Why must paired data stay matched to the same person/object?

Otherwise the relationship loses meaning.

What is a positive relationship?

As one variable increases, the other tends to increase.

What is a negative relationship?

As one variable increases, the other tends to decrease.

What does no observed relationship mean?

There is no clear pattern between the variables.

What does a strong relationship look like on a scatterplot?

Points are clustered closely around a trend.

What does a weak relationship look like on a scatterplot?

Points are more spread out.

What is a linear relationship?

A relationship that is approximately described by a straight line.

Can two variables be related but not linearly?

Yes.

What does correlation measure?

The direction and strength of a linear relationship between two variables.

What does positive correlation mean?

The variables tend to increase together.

What does negative correlation mean?

One variable tends to increase while the other decreases.

What does correlation near 0 mean?

Little to no linear relationship.

Does correlation prove causation?

No.

What is a line of best fit?

A line that summarizes the overall linear trend in the data.

General equation of a line of best fit

y = ax + b

In a line of best fit, what does slope tell you?

How much the response variable changes per unit change in the explanatory variable.

In a line of best fit, what does the intercept tell you?

The predicted value of y when x = 0.

What does it mean to predict from a line?

Use the line to estimate a likely y-value for a given x-value.

Why should you be careful predicting far outside the data range?

Because the pattern may not continue outside the observed data.

What is a residual?

The difference between the observed value and the predicted value.

Residual formula

residual = y - y-hat

What does a positive residual mean?

The actual point is above the line.

What does a negative residual mean?

The actual point is below the line.

If body temperature is being used to explain heart rate, what is the explanatory variable?

Body temperature.

If body temperature is being used to explain heart rate, what is the response variable?

Heart rate.

If sleep is used to predict reaction time, what is the explanatory variable?

Sleep.

If sleep is used to predict reaction time, what is the response variable?

Reaction time.

If a scatterplot slopes upward, the relationship is

Positive.

If a scatterplot slopes downward, the relationship is

Negative.

If points form a random cloud, the relationship is

No observed relationship.

What is the difference between explanatory and response variables?

The explanatory variable helps explain or predict the response variable.

Why is heart rate and body temperature considered paired data?

Because each heart rate must be matched to the correct person’s body temperature.

Can a relationship exist even if the points are not perfectly on a line?

Yes.

What does “trend” mean in a scatterplot?

The overall pattern the points follow.

What does “clustered” mean in a scatterplot?

The points are close together.

What does “spread out” mean in a scatterplot?

The points are more scattered and less tightly grouped.

If two variables move together upward, what kind of association is that?

Positive association.

If one variable increases while the other decreases, what kind of association is that?

Negative association.

What is the purpose of a scatterplot?

To visualize the relationship between two numerical variables.

What kind of variables are used in a scatterplot?

Two numerical variables.

Can scatterplots show causation?

No, they only show association / relationship.

Why is the mean more affected by outliers?

Because extreme values pull the average up or down.

Why is the median more resistant to outliers?

Because it depends on the middle position, not the size of extreme values.

If a few very high values are added to a data set, which changes more: mean or median?

Mean.

What does “resistant” mean in statistics?

Not strongly affected by outliers.

If a professor asks whether the relationship is positive, negative, or none, what are they asking about?

The direction of the relationship.

If a professor asks whether the relationship is strong or weak, what are they asking about?

The strength of the relationship.

If a relationship is curved rather than straight, is it linear?

No.

If correlation is strong and positive, what does the scatterplot usually look like?

Points clustered around an upward trend.

If correlation is strong and negative, what does the scatterplot usually look like?

Points clustered around a downward trend.

If correlation is close to zero, what does the scatterplot usually look like?

A random cloud with no clear linear trend.

What is the “best fit” line trying to do?

Get as close as possible to the overall pattern of the points.