The Probability of a Type II Error
and the Power of the Test
Probability of a Type II Error /
Power of Test
• Type I Error: Rejecting Null Hypothesis when
Null Hypothesis is true (α is the probability of
a Type I Error)
• Type II Error: Accepting Null Hypothesis when
Null Hypothesis is false.
Review: How to Determine Picture
H1: p ≠ value
H1: p < value
H1: p > value
Two Tails
Left Tail
Right Tail
Finding Probability of a Type II Error /
Power of Test (Proportion) (TI-83/84)
1.
2.
3.
4.
Write down claim
Write down null and alternate hypothesis
Draw picture (determine tail(s))
Find ZAL and/or ZAR (you will have both for
two tails) – round to 3 decimal places
Left Tail: ZAL = INVNORM(α)
Right Tail: ZAR = INVNORM(1-α )
Two Tails: ZAL = INVNORM(α/2)
and ZAR = INVNORM(1-α/2)
Finding Probability of a Type II Error /
Power of Test (Proportion) (TI-83/84)
Finding Probability of a Type II Error /
Power of Test (Proportion) (TI-83/84)
Finding Probability of a Type II Error /
Power of Test (Proportion) (TI-83/84)
7. Find the Probability of a Type II Error (β)
• Left Tail: β = NORMALCDF(zL,E99)
• Right Tail: β = NORMALCDF(-E99,zR)
• Two Tails: β = NORMALCDF(zL,zR)
8. Find the power of the test (1-β)
Finding Probability of a Type II Error /
Power of Test (Proportion) (By Hand)
1.
2.
3.
4.
Write down claim
Write down null and alternate hypothesis
Draw picture (determine tail(s))
Find ZAL and/or ZAR (you will have both for two
tails) – Use Standard Normal Distribution table
Left Tail: ZAL = Look up α
Right Tail: ZAR = Look up (1-α)
Two Tails: ZAL = Look up (α/2)
and ZAR = Look up (1-α/2)
Finding Probability of a Type II Error /
Power of Test (Proportion) (By Hand)
Finding Probability of a Type II Error /
Power of Test (Proportion) (By Hand)
Finding Probability of a Type II Error /
Power of Test (Proportion) (By Hand)
7. Find the Probability of a Type II Error (β). Look up z
scores in standard normal distribution table.
• Left Tail: β = 1 – (value from table based on zL)
• Right Tail: β = (value from table based on zR)
• Two Tails: β = (value from table based on zR) – (value
from table based on zL)
8. Find the power of the test (1-β)
1. Find Probability of Type II Error /
Power of Test
To test Ho: p = 0.40 versus H1: p < 0.40, a simple
random sample of n = 200 is obtained and 90
successes are observed. If the researcher
decides to test this hypothesis at the α = 0.05
level of significance, compute the probability of
making a Type II Error if the true population
proportion is 0.38. What is the power of the
test?
2. Find Probability of Type II Error /
Power of Test
To test Ho: p = 0.30 versus H1: p ≠ 0.30, a simple
random sample of n = 500 is obtained and 170
successes are observed. If the researcher
decides to test this hypothesis at the α = 0.01
level of significance, compute the probability of
making a Type II Error if the true population
proportion is 0.34. What is the power of the
test?
Finding Probability of a Type II Error /
Power of Test (Mean) (TI-83/84)
1.
2.
3.
4.
Write down claim
Write down null and alternate hypothesis
Draw picture (determine tail(s))
Find ZAL and/or ZAR (you will have both for
two tails) – round to 3 decimal places
Left Tail: ZAL = INVNORM(α)
Right Tail: ZAR = INVNORM(1-α)
Two Tails: ZAL = INVNORM(α/2)
and ZAR = INVNORM(1-α/2)
Finding Probability of a Type II Error /
Power of Test (Mean) (TI-83/84)
Finding Probability of a Type II Error /
Power of Test (Mean) (TI-83/84)
Finding Probability of a Type II Error /
Power of Test (Mean) (TI-83/84)
7. Find the Probability of a Type II Error (β)
• Left Tail: β = NORMALCDF(zL,E99)
• Right Tail: β = NORMALCDF(-E99,zR)
• Two Tails: β = NORMALCDF(zL,zR)
8. Find the power of the test (1-β)
Finding Probability of a Type II Error /
Power of Test (Mean) (By Hand)
1.
2.
3.
4.
Write down claim
Write down null and alternate hypothesis
Draw picture (determine tail(s))
Find ZAL and/or ZAR (you will have both for
two tails) – Use Standard Normal Distribution
table
Left Tail: ZAL = Look up α
Right Tail: ZAR = Look up (1-α)
Two Tails: ZAL = Look up (α/2)
and ZAR = Look up (1-α/2)
Finding Probability of a Type II Error /
Power of Test (Mean) (By Hand)
Finding Probability of a Type II Error /
Power of Test (Mean) (By Hand)
Finding Probability of a Type II Error /
Power of Test (Mean) (By Hand)
7. Find the Probability of a Type II Error (β). Look up z
scores in standard normal distribution table.
• Left Tail: β = 1 – (value from table based on zL)
• Right Tail: β = (value from table based on zR)
• Two Tails: β = (value from table based on zR) – (value
from table based on zL)
8. Find the power of the test (1-β)
3. Find Probability of Type II Error /
Power of Test
To test Ho: μ = 400 versus H1: μ > 400, a simple
random sample of n = 100 is obtained. Assume
the population standard deviation is 80. If the
researcher decides to test this hypothesis at the
α = 0.05 level of significance, compute the
probability of making a Type II Error if the true
population mean is 420. What is the power of
the test?