15 Hypothesis Testing Part 1

# Hypothesis testing computing the p value 15 h0 μ μ0

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Hypothesis testing: Computing the p-value 15

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H0: μ ≥(=) μ0 HA: μ < μ0 0 (1) Lower tail test 0 (2) Upper tail test H0: μ ≤(=) μ0 HA: μ > μ0 extrem e extrem e (3) Two-tailed test extrem e extrem e 0 H0: μ = μ0 HA: μ ≠ μ0 P(T < t) P(T > t) 2 *P(T > |t|) Always compute p-value in direction of alternative hyp. 16
Hypothesis testing: P-value steps 1. Use sample data to compute sample statistic (for example, sample mean or sample proportion) 2. Compute “test statistic”: z or t (= number of standard deviations from the mean your sample statistic is) 3. Compute p-value for test statistic: The probability of getting that test statistic or one more extreme (in the direction of the alternative hypothesis) if the null is true 4. Compare p-value to α 1. Reject Ho if p-value is < α (met the burden of proof) 2. Do not reject Ho: p-value is > α (didn’t meet the burden of proof) 17

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General form of test statistic: As s ume the null is true Hypothesis testing: Computing test statistics 18 on distributi sampling of error standard value parameter null statistic sample t) (or z - =
Test Statistics for Population Parameters (Large samples) Population proportion, p (large sample: p*n > 5 and (1-p)*n > 5) Parameter Test Statistic Standard Error Population mean, µ: σ Unknown (large sample: n ≥ 30) Computing test statistics: General formulas 19 Population mean, µ: σ Known (large sample: n ≥ 30) n s s x x = n p p p ) 1 ( 0 0 ˆ - = σ p p p z ˆ 0 ˆ σ - = x s x t 0 μ - = σ x = σ n z = x - μ 0 σ x

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Want to show µ < 50; α=0.05 1. Hypotheses: H0: µ ≥ 50 HA: µ < 50 2. Data: Sample x̅ = 25 σ = 30 n = 36 3. Test statistic: extrem e 0 Z z = -5.0 µ 0 = 50 x̅ = 25 5. Conclusion: 0.0000<0.05 Reject the null in favor of the alternative: conclude that the mean age of Cadillac buyers is younger than 50! 4. P-value P(Z < -5.0) = 0.0000 Cadillac example revisited 21 0 . 5 36 30 50 25 . 0 0 - = - = - = - = n x error std x z x σ μ μ

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What if our hypotheses had been different? Now suppose we want to show that the mean age of Cadillac buyers is older than 50? Our hypotheses: H0: µ ≤ 50 HA: µ > 50 Sampling distribution of mean is normal Get a sample mean of 25 We said that 25 is quite odd if Cadillac buyers are 50 years old or more, but is it odd if they are younger? Does it support our alternative that Cadillac buyers are older than 50? Cadillac example revisited 22
Want to show µ > 50; α=0.05 1. Hypotheses: H0: µ ≤ 50 HA: µ > 50 2. Data: Sample x̅ = 25 σ = 30 n = 36 3. Test statistic: extrem e 0 Z z = -5.0 µ 0 = 50 x̅ = 25 4. P-value P(Z > -5.000)= 1.000 5. Conclusion: 1.0000 < .05 Fail to reject the null; find no evidence that the mean age of Cadillac buyers is older than 50!

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