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Hypothesis Testing Errors Hypothesis Testing • Suppose we believe the average systolic blood pressure of healthy adults is normally distributed with mean μ = 120 and variance σ2 = 50. • To test this assumption, we sample the blood pressure of 42 randomly selected adults. Sample statistics are Mean X = 122.4 Variance s2 = 50.3 Standard Deviation s = √50.3 = 7.09 Standard Error = s / √n = 7.09 / √42 = 1.09 Z0 = ( X – μ ) / (s / √n) = (122.4 – 120) / 1.09 = 2.20 Confidence Interval 95% Level of Significance a = 5% 95% a / 2 = 2.5% a / 2 = 2.5% Z0 = 2.20 -Za/2 = -1.96 +Za/2 = +1.96 Conclusion (Critical Value) Since Z0= 2.20 exceeds Zα/2 = 1.96, Reject H0: μ = 120 and Accept H1: μ ≠ 120. Conclusion (p-Value) We can quantify the probability (p-Value) of obtaining a test statistic Z0 at least as large as our sample Z0. P( |Z0| > Z ) = 2[1- Φ (|Z0|)] p-Value = P( |2.20| > Z ) = 2[1- Φ (2.20)] p-Value = 2(1 – 0.9861) = 0.0278 = 2.8% Compare p-Value to Level of Significance If p-Value < α, then reject null hypothesis Since 2.8% < 5%, Reject H0: μ = 120 and conclude μ ≠ 120. Confidence Interval = 99% Level of Significance α = 1% Z0 = ( X – μ ) / (s / √n) = (122.4 – 120) / 1.09 = 2.20 Zα/2 = +2.58 Confidence Interval 99% Level of Significance a = 1% 99% a / 2 = 0.5% a / 2 = 0.5% Z0 = 2.20 -Za/2 = -2.58 +Za/2 = +2.58 Conclusion (Critical Value) Since Z0= 2.20 is less than Zα/2 =2.58, Fail to Reject H0: μ = 120 and conclude there is insufficient evidence to say H1: μ ≠ 120. Conclusion (p-Value) We can quantify the probability (p-Value) of obtaining a test statistic Z0 at least as large as our sample Z0. P( |Z0| > Z ) = 2[1- Φ (|Z0|)] p-Value = P( |2.20| > Z ) = 2[1- Φ (2.20)] p-Value = 2(1 – 0.9861) = 0.0278 = 2.8% Compare p-Value to Level of Significance If p-Value < α, then reject null hypothesis Since 2.8% > 1%, Fail to Reject H0: μ = 120 and conclude there is insufficient evidence to say H1: μ ≠ 120. Hypothesis Testing Conclusions • As can be seen in the previous example, our conclusions regarding the null and alternate hypotheses are dependent upon the sample data and the level of significance. • Given different values of sample mean and the sample variance or given a different level of significance, we may come to a different conclusion.