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1. The regression model and its conditions

Warm-up

Three quick questions from earlier lessons.

Discover

Two random samples of 30 students give regression slopes 2.1 and 2.6 for hours studied vs score. Why do they differ?

Sampling variability: each sample estimates the same true slope
One of the samples must be wrong

Key idea

A model for the population.

Population regression line:

\mu_y = \beta_0 + \beta_1 x

, estimated by

\hat{y} = a + bx

. Conditions for inference (LINER): Linear relationship, Independent observations, Normal residuals, Equal spread of residuals, Random sample or assignment.

Worked example

Checking the conditions with a residual plot.

Linear: residuals scattered around 0

No curve.

Equal spread across

x

No fan.

Roughly normal: no strong skew or outliers

Histogram of residuals.

Your turn, guided

Your turn: how many conditions are in LINER?

5

L, I, N, E, R.

Practice ★

A regression on a random sample of 23 houses gives slope b. What does b estimate?
1. The slope β₁ of the population regression line
2. Nothing: b is the true slope
3. The correlation r

Which conditions are needed for inference about the slope from 75 pairs? Tick all that are.
1. The true relationship is linear
2. Observations are independent (random sample, 10% condition)
3. Residuals are roughly normal
4. Residuals have about equal spread for all x
5. The correlation is above 0.5

Watch out!

Practice ★★

The residual plot of a regression on 85 points fans out: residuals spread more as x grows. Which condition fails?
1. Equal variance of the residuals
2. Linearity
3. Normality of the population

Exit ticket

Three questions to check you have got it.

A regression on a random sample of 23 houses gives slope b. What does b estimate?
1. The slope β₁ of the population regression line
2. Nothing: b is the true slope
3. The correlation r
A regression on a random sample of 59 houses gives slope b. What does b estimate?
1. The slope β₁ of the population regression line
2. Nothing: b is the true slope
3. The correlation r
A regression on a random sample of 46 houses gives slope b. What does b estimate?
1. The slope β₁ of the population regression line
2. Nothing: b is the true slope
3. The correlation r

Summary card

The regression model

$\hat{y} = a + bx$ estimates $\mu_y = \beta_0 + \beta_1 x$ . Conditions: LINER.

Watch out: no condition on the size of $r$ .

2. The t statistic for the slope

Warm-up

Three quick questions from earlier lessons.

Discover

A sample slope is b = 0.6 with standard error 0.2. How many standard errors is it from 0?

3
0.12

Key idea

Standardize the slope.

t = \dfrac{b - \beta_1^{(0)}}{SE_b}

, usually with

\beta_1^{(0)} = 0

, and df = $n - 2$ . Computer output lists

b

(Coef),

SE_b

(SE Coef),

t

and the P-value in the row of

x

Worked example

Output for $x$ : Coef 1.25, SE Coef 0.50, $n = 22$ .

t = 2.5

1.25/0.50.

= 20

22 − 2.

P-value

\approx 0.021

Two-sided, from the t distribution.

Your turn, guided

Your turn: b = 0.9, SE_b = 0.3. Find t.

t = 3

0.9/0.3.

Practice ★

Computer output gives slope b = 8/10 with standard error SE_b = 3/100. Find the t statistic for H₀: β₁ = 0.
Computer output gives slope b = 9/10 with standard error SE_b = 8/100. Find the t statistic for H₀: β₁ = 0.

A regression uses 41 data pairs. How many degrees of freedom does the t test for the slope have?

Watch out!

Practice ★★

In regression output, the row for x reads: Coef 0.97, SE Coef 0.08, and a t value. Which number is the slope's standard error?
1. The SE Coef in the x row
2. The Coef in the x row
3. The value s (the SD of the residuals)

Boss problem. A regression on 30 pairs gives b = 9/10 and SE_b = 7/100.

Exit ticket

Three questions to check you have got it.

Computer output gives slope b = 8/10 with standard error SE_b = 3/100. Find the t statistic for H₀: β₁ = 0.
Computer output gives slope b = 9/10 with standard error SE_b = 8/100. Find the t statistic for H₀: β₁ = 0.
Computer output gives slope b = 7/10 with standard error SE_b = 3/100. Find the t statistic for H₀: β₁ = 0.

Summary card

t for the slope

$t = b/SE_b$ , df $= n - 2$ .

Watch out: divide by $SE_b$ ; two parameters, so $n - 2$ .

3. Confidence interval for the slope

Warm-up

Three quick questions from earlier lessons.

Discover

b = 0.5, SE_b = 0.1, t* = 2. Which interval estimates β₁?

(0.3, 0.7)
(0.4, 0.6)

Key idea

An interval for the population slope.

b \pm t^* \cdot SE_b

, with

t^*

from the t distribution with

n - 2

df. If the interval does not contain 0, there is convincing evidence of a linear relationship; interpret as the change in mean

y

per unit increase in

x

Worked example

$n = 30$ , $b = 0.42$ , $SE_b = 0.10$ , $t^* = 2.048$ .

ME = 0.2048

2.048 × 0.10.

(0.215, 0.625)

0.42 ± 0.2048.

Evidence of a positive slope

0 excluded.

Your turn, guided

Your turn: ME = 0.15. The width of the interval.

0.3

2 × 0.15.

Practice ★

With 30 pairs, b = 2/10 and SE_b = 7/100. Using t* = 2.048 (28 df, 95%), find the upper end of the interval for β₁.
With 30 pairs, b = 4/10 and SE_b = 1/100. Using t* = 2.048 (28 df, 95%), find the upper end of the interval for β₁.

SE_b = 1/100 and t* = 2.048. Find the width of the 95% confidence interval for β₁.

Watch out!

Practice ★★

A 95% interval for the slope β₁ is (0.4, 0.8). What does it suggest?
1. Convincing evidence of a positive linear relationship: 0 is not in the interval
2. No evidence of a relationship
3. x causes y to increase

Exit ticket

Three questions to check you have got it.

With 30 pairs, b = 2/10 and SE_b = 7/100. Using t* = 2.048 (28 df, 95%), find the upper end of the interval for β₁.
With 30 pairs, b = 4/10 and SE_b = 1/100. Using t* = 2.048 (28 df, 95%), find the upper end of the interval for β₁.
With 30 pairs, b = 3/10 and SE_b = 6/100. Using t* = 2.048 (28 df, 95%), find the upper end of the interval for β₁.

Summary card

Interval for β₁

$b \pm t^* SE_b$ (df $n - 2$ ); 0 outside → evidence of a relationship.

Watch out: association, not causation.

4. Testing the slope and reading output

Warm-up

Three quick questions from earlier lessons.

Discover

If x is no help in predicting y, what is the population slope?

Key idea

Test whether the slope is zero.

H_0: \beta_1 = 0

(no linear relationship),

H_a: \beta_1 \ne 0

(or one-sided). Use

t = b/SE_b

, df

= n - 2

. A small P-value is convincing evidence of a linear relationship in the population. Read

b

SE_b

t

and P from the output.

Worked example

Output: b = 0.30, t = 6.0, P < 0.001.

SE_b = 0.30/6 = 0.05

t = b/SE_b.

Reject H₀

Very small P.

Convincing evidence of a linear relationship

In context.

Your turn, guided

Your turn: b = 0.8 and t = 4. Find $SE_b$ .

0.2

0.8/4.

Practice ★

To test whether x is useful for predicting y (from 51 pairs), what is the null hypothesis?
1. H₀: β₁ = 0, no linear relationship in the population
2. H₀: b = 0
3. H₀: r = 1

Output shows slope b = 6/10 and t = 9 for H₀: β₁ = 0. Find SE_b.
Output shows slope b = 3/10 and t = 9 for H₀: β₁ = 0. Find SE_b.

Watch out!

Practice ★★

An observational study tests H₀: β₁ = 0 for hours of sleep (x) and test score (y). The P-value is 0.006. What can you conclude?
1. Convincing evidence of a linear association between sleep and scores in the population sampled
2. More sleep causes higher scores
3. The correlation must be close to 1

Exit ticket

Three questions to check you have got it.

Output shows slope b = 6/10 and t = 9 for H₀: β₁ = 0. Find SE_b.
Output shows slope b = 3/10 and t = 9 for H₀: β₁ = 0. Find SE_b.
Output shows slope b = 3/10 and t = 6 for H₀: β₁ = 0. Find SE_b.

Summary card

Testing the slope

H₀: β₁ = 0; $t = b/SE_b$ , df $= n - 2$ .

Watch out: a significant slope is not proof of causation.

Answers: Inference for regression

1. The regression model and its conditions

The slope β₁ of the population regression line

The true relationship is linear · Observations are independent (random sample, 10% condition) · Residuals are roughly normal · Residuals have about equal spread for all x

Equal variance of the residuals

The slope β₁ of the population regression line
The slope β₁ of the population regression line
The slope β₁ of the population regression line

2. The t statistic for the slope

$\frac{80}{3}$
$\frac{45}{4}$

$39$

The SE Coef in the x row

$\frac{4729}{6250}$

$\frac{80}{3}$
$\frac{45}{4}$
$\frac{70}{3}$

3. Confidence interval for the slope

$\frac{1073}{3125}$
$\frac{1314}{3125}$

$\frac{128}{3125}$

Convincing evidence of a positive linear relationship: 0 is not in the interval

$\frac{1073}{3125}$
$\frac{1314}{3125}$
$\frac{2643}{6250}$

4. Testing the slope and reading output

H₀: β₁ = 0, no linear relationship in the population

$\frac{1}{15}$
$\frac{1}{30}$

Convincing evidence of a linear association between sleep and scores in the population sampled

$\frac{1}{15}$
$\frac{1}{30}$
$\frac{1}{20}$