Unit 9: Inference for Slopes

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Last updated 2:44 PM on 4/15/26
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4 Terms

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Least Square Regression Line

It is the line that minimizes the sum of squared residuals → y​=a+bx

  • Slope (b) - How much y changes when x increases by 1

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Computer Output

  • The p-value is only for a two sided test → don’t forget to divide by two for a one sided test!

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Confidence Interval for Slopes

A confidence interval for the slope tells you a plausible range for the true population slope β.

1) STATE

  • Parameter: Let β be the true slope of the population LSRL for (x) and (y)

  • Statistic (b) = _ → fond from computer output

  • Confidence level

2) PLAN (L.I.N.E.R): t Interval for Slopes

  • Linear - The dot plot shows a linear regression and the residual plot has no pattern

  • Independent - The sample size < 10% of population size

  • Normal - The residual plot has no skew or outliers → Use normal distribution

  • Equal Standard Deviation - The residual plot shows a similar variability of x

  • Random - They took a random sample → Establish causation

3) DO t Interval for Slopes

Confidence Interval: b±t(SEb)

  • b → statistic (found from computer output)

  • t → invT( (1-confidence level)/2, df (n-2)

  • SEb → found from computer output

4) Conclude

We are _% confident that the interval _ to _ captures the true population slope of the LSRL for (x) and (y)

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Hypothesis Tests for Slopes

1) STATE

  • Parameter: Let β be the true slope of the population LSRL for (x) and (y)

  • Hypothesis test

    • H0: β = 0

    • Ha: β <,>, ≠ 0

  • Statistic (b) = _ → fond from computer output

  • Significance level

2) PLAN (L.I.N.E.R) t Test for Slopes

  • Linear - The dot plot shows a linear regression and the residual plot has no pattern

  • Independent - The sample size < 10% of population size

  • Normal - The residual plot has no skew or outliers → Use normal distribution

  • Equal Standard Deviation - The residual plot shows a similar variability of x

  • Random - They took a random sample → Establish causation

3) DO

  • t Statistic : (b-β) / SEb → Found from computer output

  • p value: Found from computer output

    • df = n-2

4) CONCLUDE

If p < 0.05 → Reject null

If p > 0.05 → Fail to reject null