Testing the slope and reading output

About this lesson

Statistics (AP-aligned) › Inference for regression

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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?

  1. 0
  2. 1

Key idea

Test whether the slope is zero.

H0:β1=0H_0: \beta_1 = 0 (no linear relationship), Ha:β1≠0H_a: \beta_1 \ne 0 (or one-sided). Use t=b/SEbt = b/SE_b, df =n−2= n - 2. A small P-value is convincing evidence of a linear relationship in the population. Read bb, SEbSE_b, tt and P from the output.

Worked example

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

SEb=0.30/6=0.05SE_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 SEbSE_b.

0.20.2
0.8/4.

Practice ★

  1. 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
  1. Output shows slope b = 6/10 and t = 9 for H₀: β₁ = 0. Find SE_b.
  2. Output shows slope b = 3/10 and t = 9 for H₀: β₁ = 0. Find SE_b.

Watch out!

Practice ★★

  1. 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.

  1. Output shows slope b = 6/10 and t = 9 for H₀: β₁ = 0. Find SE_b.
  2. Output shows slope b = 3/10 and t = 9 for H₀: β₁ = 0. Find SE_b.
  3. 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/SEbt = b/SE_b, df =n−2= n - 2.

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

End of the lesson

Back to the chapter

Answers: Testing the slope and reading output

A

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

B

  1. 115\frac{1}{15}
  2. 130\frac{1}{30}

C

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

D

  1. 115\frac{1}{15}
  2. 130\frac{1}{30}
  3. 120\frac{1}{20}

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