Transcript Hypothesis Testing Type I & Type II Errors

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.