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 σ 2 = 50. To test this assumption, we sample the blood pressure of 42 randomly selected adults. Sample statistics are Mean X = 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

 / 2  5% Confidence Interval 95% Level of Significance  5%       95% Z 0 = 2.20

Conclusion (Critical Value) Since Z 0 = 2.20 exceeds Z α/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 – ) = = 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 Z α/2 = +2.58

 / 2  5% Confidence Interval 99% Level of Significance  1%       99% Z 0 = 2.20

Conclusion (Critical Value) Since Z 0 = 2.20 is less than Z α/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 – ) = = 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.