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 Two Tails H1: p < value Left Tail H1: p > value Right Tail

Finding Probability of a Type II Error / Power of Test (Proportion) (TI-83/84) 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) 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) 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) 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?