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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind STAT 113 Tests and Confidence Intervals Colin Reimer Dawson Oberlin College October 10th, 2016 Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind Reminders and

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

STAT 113 Tests and Confidence Intervals

Colin Reimer Dawson

Oberlin College

October 10th, 2016

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Reminders and Announcements

  • HW online, due Friday (but ok if you want to turn it in during

break)

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Two-Tailed Tests

Two-Tailed Test

In a Two-Tailed Test, H1 does not specify the direction (sign) of a difference/correlation/slope. So outcomes at either extreme count in its favor. The P-value therefore uses outcomes at or past the

  • bserved one, but also the symmetric outcomes on the other “tail”

We should prefer two-tailed tests, unless only one side of the alternative is plausible a priori.

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

What is low enough?

Significance level (α)

We need to decide for ourselves, in advance of collecting data, what we will count as a “low enough” P-value to achieve statistical

  • significance. This threshold is called the significance level of the
  • test. (Notation: α)
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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Making a Decision

Reject H0 or not?

Compare P to α. (a) P ≥ α: Do not reject H0. (Data wouldn’t be that surprising if H0 true. H0 is “presumed innocent”.) (b) P < α: Reject H0. (Data would be too surprising if H0 were

  • true. Beyond a “reasonable doubt”.)

We do not “accept H0”. We “fail to reject” it. (Not enough evidence to decide)

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Types of Errors

2 × 2 table of possibilities. Is H0 actually false (does the treatment actually work)? Did we reject H0 (did we conclude that it works)? Action H0 rejected H0 not rejected Truth H0 is false True Discovery Missed Discovery H0 is true False Discovery No Error

Table: Possible outcomes of a null hypothesis significance test

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

  • We can set α to whatever we want. The lower it is, the less
  • ften we make Type I Errors.
  • Tradeoff: Fewer Type I Errors → More Type II Errors.
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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

Decreasing α moves the rejection threshold out toward the tail of the H0 distribution.

5 10 15 20 0.00 0.05 0.10 0.15 0.20 Values Probability

  • α = 0.15, threshold = 8

Blue spikes: Distribution of outcomes if H0 is true

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

Decreasing α moves the rejection threshold out toward the tail of the H0 distribution.

5 10 15 20 0.00 0.05 0.10 0.15 0.20 Values Probability

  • α = 0.05, threshold = 9

Blue spikes: Distribution of outcomes if H0 is true

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

Decreasing α moves the rejection threshold out toward the tail of the H0 distribution.

5 10 15 20 0.00 0.05 0.10 0.15 0.20 Values Probability

  • α = 0.01, threshold = 11

Blue spikes: Distribution of outcomes if H0 is true

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

We retain H0 when we do not exceed the threshold. But if H1 is correct, this is a Type II Error. More stringent threshold → missed discoveries.

5 10 15 20 0.00 0.05 0.10 0.15 0.20 Values Probability

  • α = 0.15, threshold = 8

Blue spikes: Distribution of outcomes if H0 is true Orange spikes: Distribution of outcomes for one possible parameter value under .

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

We retain H0 when we do not exceed the threshold. But if H1 is correct, this is a Type II Error. More stringent threshold → missed discoveries.

5 10 15 20 0.00 0.05 0.10 0.15 0.20 Values Probability

  • α = 0.05, threshold = 9

Blue spikes: Distribution of outcomes if H0 is true Orange spikes: Distribution of outcomes for one possible parameter value under .

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Type I vs. Type II Errors

We retain H0 when we do not exceed the threshold. But if H1 is correct, this is a Type II Error. More stringent threshold → missed discoveries.

5 10 15 20 0.00 0.05 0.10 0.15 0.20 Values Probability

  • α = 0.01, threshold = 11

Blue spikes: Distribution of outcomes if H0 is true Orange spikes: Distribution of outcomes for one possible parameter value under .

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Two-Tailed Tests and Stat. Significance Worksheet: Love is Blind

Worksheet: Love is Blind, Continued