Inference for Regression Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inference for Regression.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Using hypothesis tests and confidence intervals to draw conclusions about the true population slope $\beta_1$ of the linear relationship $y = \beta_0 + \beta_1 x + \varepsilon$ , based on sample data.

You computed a sample regression line with slope $b = 2.3$ . But is the true population slope actually different from zero? Maybe there's really no linear relationship and you just got a slope by chance. The regression t-test asks: 'Is my sample slope far enough from zero that it's unlikely to have occurred by random variation alone?'

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Inference for regression tests whether the population slope $\beta_1$ differs from zero using a t-test on the sample slope.

Common stuck point: The procedure for inference for regression is the easy part; the trap is treating a nonzero sample slope as proof of a population relationship. Asking "Am I testing whether the underlying population slope is nonzero (rather than just computing or describing the sample slope)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I testing whether the underlying population slope is nonzero (rather than just computing or describing the sample slope)?

Worked Examples

Example 1

medium

A regression output shows: slope

b=2.5

SE_b=0.8

n=30

. Test

H_0: \beta=0

H_a: \beta \neq 0

\alpha=0.05

using a t-test.

Answer

t=3.125 > 2.048

. Reject

H_0

. The slope is statistically significant at

\alpha=0.05

First step

Test statistic:

t = \frac{b - \beta_0}{SE_b} = \frac{2.5 - 0}{0.8} = 3.125

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Example 2

hard

Construct a 95% confidence interval for the slope

\beta

given:

b=1.8

SE_b=0.5

n=25

, and

t^*_{0.025,23}=2.069

Example 3

medium

A study gives

b = 3.2

SE_b = 1.0

n = 12

. With

t^*_{0.025, 10} = 2.228

, construct the 95% CI for the slope.

Example 4

medium

Why is the slope SE inversely related to the spread of the

x

values?

Example 5

hard

Explain why a significant slope does NOT imply a causal relationship between

x

and

y

Example 6

medium

A scatterplot shows a clear curved pattern, but a linear regression gives slope p-value

< 0.001

. What is the danger of trusting this inference?

Example 7

hard

A regression has

b = 1.0

SE_b = 0.5

n = 25

. A 99% CI uses

t^*_{0.005, 23} = 2.807

. Compute the CI.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

List the four conditions for valid regression inference and explain why each must be checked.

Example 2

hard

A regression of salary on years of experience gives:

\hat{y} = 30000 + 2000x

R^2=0.72

, slope p-value=0.001. A confidence interval for the slope is

(1500, 2500)

. Provide a full interpretation of each result.

Example 3

easy

In regression inference with sample size

n = 20

, how many degrees of freedom does the t-test for the slope use?

Example 4

easy

What is the null hypothesis when testing whether a linear relationship exists between

x

and

y

Example 5

easy

The slope t-statistic is

t = \frac{b}{SE_b}

. If

b = 2.4

and

SE_b = 0.6

, compute

t

Example 6

easy

A slope test gives a p-value of

0.002

\alpha = 0.05

. What is the conclusion about the slope?

Example 7

easy

A confidence interval for the slope is

(1.2, 3.8)

. Does it provide evidence of a nonzero slope at the matching significance level?

Example 8

easy

A slope CI of

(-0.5, 2.0)

is reported. What does it suggest about the slope being zero?

Example 9

easy

Name one condition that must hold for regression inference to be valid.

Example 10

easy

A significant positive slope is found between ice cream sales and drowning deaths. Does this prove ice cream causes drownings?

Example 11

medium

Compute the slope t-statistic and state the df:

b = 1.5

SE_b = 0.5

n = 12

Example 12

medium

Build a slope CI:

b = 4.0

SE_b = 0.8

t^* = 2.0

. Use

b \pm t^* SE_b

Example 13

medium

A slope test yields

t = 1.2

df = 15

, p-value

= 0.25

, at

\alpha = 0.05

. Conclude.

Example 14

medium

A residual plot for a regression shows a clear curve. Why does this matter before doing slope inference?

Example 15

medium

The standard error of the slope is

SE_b = \frac{s}{s_x \sqrt{n-1}}

. If

s = 6

s_x = 3

n = 26

, compute

SE_b

Example 16

medium

A two-sided slope test has

t = 2.5

df = 8

. For a one-sided test (slope

> 0

) with the same data, how does the p-value change?

Example 17

medium

A slope estimate is

b = 0.0

exactly with

SE_b = 0.5

. What is the slope t-statistic and likely conclusion?

Example 18

medium

A 95% slope CI is

(0.0, 4.0)

. At

\alpha = 0.05

(two-sided), what is the conclusion about

\beta_1 = 0

Example 19

medium

A slope CI is built as

b \pm t^* SE_b

with

b = 5

t^* = 2.0

SE_b = 1.0

. Does the resulting interval show a significant slope?

Example 20

challenge

A regression on

n = 27

points has

b = 3.0

and

SE_b = 1.0

. Find

t

, the df, and (using

t^* = 2.06

) a 95% CI for the slope.

Example 21

challenge

An experiment (random assignment) finds a significant positive slope between dose and response. A separate observational study finds the same slope significant. Which study can support a causal claim, and why?

Example 22

challenge

A slope test has

df = n - 2 = 8

. A student mistakenly used

df = n - 1 = 9

, getting a slightly different p-value. Explain the correct df and why regression loses one more degree of freedom than a one-sample mean.

Example 23

easy

A regression has

b = 0.8

and

SE_b = 0.2

. Compute the slope

t

-statistic.

Example 24

easy

A 95% CI for slope is

(0.4, 1.6)

. Is the slope significantly different from zero at

\alpha = 0.05

Example 25

medium

A regression has

b = 0.05

with a p-value of

0.62

. What is the conclusion at

\alpha = 0.05

Example 26

medium

For

b = 1.5

SE_b = 0.5

n = 20

, find the 95% CI (

t^*_{0.025, 18} = 2.101

Example 27

medium

A regression analysis reports

t = 2.5

with

df = 22

. Approximately, is this significant at

\alpha = 0.05

(two-sided)?

Example 28

hard

A small sample regression gives

b = 4.0

SE_b = 1.2

n = 8

. Test

H_0: \beta = 0

H_a: \beta \ne 0

\alpha = 0.05

(

t^*_{0.025, 6} = 2.447

Example 29

hard

A linear regression is fit on bivariate data; software output: slope

b = 0.15

, p-value

= 0.20

R^2 = 0.10

n = 100

. State whether the slope is statistically significant at

\alpha = 0.05

and interpret.

Example 30

hard

A 90% CI for slope is

(-0.1, 0.5)

. What is the conclusion at

\alpha = 0.10

two-sided?

Example 31

medium

A regression of test scores on study hours gives

b = 5.0

SE_b = 1.0

n = 25

. Construct a 95% CI (

t^*_{0.025, 23} = 2.069

) and interpret in context.

Example 32

medium

t = -3.0

df = 18

, find the two-sided p-value (approximate).

Example 33

easy

b = 0

exactly, what is the slope

t

-statistic?

Example 34

hard

Given

b = 2.4

SE_b = 0.8

n = 30

, test

H_0: \beta = 1.0

H_a: \beta > 1.0

(

t^*_{0.05, 28} = 1.701

Example 35

medium

A regression of height on age gives

b = 1.2

cm/yr,

SE_b = 0.4

n = 16

. Find

t

and the approximate p-value.

Example 36

challenge

A regression on a population shows true slope

\beta = 2.0

. A sample of

n = 20

gives

b = 0.5

SE_b = 1.0

. Will a 95% CI capture

\beta = 2.0

? (Use

t^* = 2.093

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Related Concepts

Least Squares Regression Line Residuals Coefficient of Determination Hypothesis Testing Confidence Interval

Background Knowledge

These ideas may be useful before you work through the harder examples.

linear regression lsrlresidualsr squaredhypothesis testingconfidence interval