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.

df

= 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 $n - 2$ .

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