Hypothesis Testing Errors Hypothesis Testing Suppose we believe
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 s 2 = 50. 3 Standard Deviation s = √ 50. 3 = 7. 09 Standard Error = s / √n = 7. 09 / √ 42 = 1. 09 Z 0 = ( X – μ ) / (s / √n) = (122. 4 – 120) / 1. 09 = 2. 20
Confidence Interval 95% Level of Significance a = 5% 95% a / 2 = 2. 5% Z 0 = 2. 20 -Za/2 = -1. 96 +Za/2 = +1. 96
Conclusion (Critical Value) Since Z 0= 2. 20 exceeds Zα/2 = 1. 96, Reject H 0: μ = 120 and Accept H 1: μ ≠ 120.
Conclusion (p-Value) We can quantify the probability (p-Value) of obtaining a test statistic Z 0 at least as large as our sample Z 0. P( |Z 0| > Z ) = 2[1 - Φ (|Z 0|)] 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 H 0: μ = 120 and conclude μ ≠ 120.
Confidence Interval = 99% Level of Significance α = 1% Z 0 = ( 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% Z 0 = 2. 20 -Za/2 = -2. 58 +Za/2 = +2. 58
Conclusion (Critical Value) Since Z 0= 2. 20 is less than Zα/2 =2. 58, Fail to Reject H 0: μ = 120 and conclude there is insufficient evidence to say H 1: μ ≠ 120.
Conclusion (p-Value) We can quantify the probability (p-Value) of obtaining a test statistic Z 0 at least as large as our sample Z 0. P( |Z 0| > Z ) = 2[1 - Φ (|Z 0|)] 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 H 0: μ = 120 and conclude there is insufficient evidence to say H 1: 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.
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