Hypothesis Testing Elements of a Hypothesis Test 1

Hypothesis Testing Elements of a Hypothesis Test: 1) Null Hypothesis H 0: µ = 21 yr 2) Alternative Hypothesis HA: µ ≠ 21 yr 3) Test Statistic X 4) Decision Rule – Rejection Region R: X > 22 yr X < 20 yr

Hypothesis Test as a Decision Table: Null Hypothesis True Accept H 0 Reject H 0 α = P(Type I Error) = P(Reject H 0|H 0 is True) β = P(Type II Error) = P(Accept H 0|H 0 is False) False

Analogy with US Criminal Legal System Person Not Guilty Acquit Convict Guilty

Use alpha to Determine the two Critical Values α/2 µ = 21 Reject H 0 Accept H 0 Standard Normal Curve Reject H 0

Hypothesis Test: H 0: µ =21 HA: µ ≠ 21 R: X > 21. 98 X < 20. 02 We could also base the test upon the Standard Normal Curve: H 0: µ = 21 HA: µ ≠ 21 α =. 05 R: Z > 1. 96 Z < -1. 96

Average Student Credit Load n =36 H 0: µ = 15 cr HA: µ ≠ 15 cr α =. 10 R: Z > Z< P-Value for a Test - Probability that H 0 is True If P-Value < α, then Reject H 0 If P-Value ≥ α, then Accept H 0 X = 14. 2 s=2

Upper Tail Test: H 0 : µ ≤ µ 0 HA : µ > µ 0 R: Z > Zα Coffee shop currently sells 320 cups/day H 0: µ ≤ 320 cups/day HA: µ > 320 cups/day α =. 05 R: Z > n =40 X = 330 cups/day s = 40 cups/day

Lower Tail Test: H 0 : µ ≥ µ 0 HA : µ < µ 0 R: Z < -Zα Employee Absenteeism currently 10. 2 day/yr H 0: µ ≥ 10. 2 day/yr HA: µ < 10. 2 day/yr α =. 10 R: Z < n =50 X = 9. 3 day/yr s = 4 day/yr

Small Sample Tests – ( n < 30 ) H 0 : µ = µ 0 HA : µ ≠ µ 0 R: t > tα/2, df=n-1 t < -tα/2, df=n-1 Hour Glass Factory: H 0: µ = 60 min HA: µ ≠ 60 min α =. 05 R: t > t< n = 25 X = 61 min s = 2 min

Work-Study Hours: n = 16 X = 11 hr/wk s = 3 hr/wk H 0: µ = 10 hr/wk HA: µ > 10 hr/wk α =. 05 R: t > 20 lb Bags of Dog Food H 0: µ = 20 lb HA: µ < 20 lb α =. 05 R: t < n = 25 X = 19 lb s = 2 lb

Tests on the Population Proportion: H 0 : p = p 0 HA : p ≠ p 0 R: Z > Zα/2 Z < -Zα/2 A Fair Coin? H 0: p =. 50 HA: p ≠. 50 α =. 05 R: Z > Z< n = 200 X = 83

Type II Error – β = P(Accept H 0|H 0 is False) Reject H 0 Accept H 0 20. 02 Suppose µ = 23 21 21. 98

1 – β = Power of the Test α = Significance of the Test µ P(Accept H 0) P(Reject H 0) 21 . 95 . 05 21. 5 . 8315 . 1685 22 . 4840 . 5160 22. 5 . 1492 . 8508 23 . 0207 . 9793