The Probability of a Type II Error and

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 H 1: p ≠ value Two Tails H 1: p < value Left Tail H 1: p > value 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) 7. Find the Probability of a Type II Error (β) • Left Tail: β = NORMALCDF(z. L, E 99) • Right Tail: β = NORMALCDF(-E 99, z. R) • Two Tails: β = NORMALCDF(z. L, z. R) 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) 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 z. L) • Right Tail: β = (value from table based on z. R) • Two Tails: β = (value from table based on z. R) – (value from table based on z. L) 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 H 1: 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 H 1: 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) 7. Find the Probability of a Type II Error (β) • Left Tail: β = NORMALCDF(z. L, E 99) • Right Tail: β = NORMALCDF(-E 99, z. R) • Two Tails: β = NORMALCDF(z. L, z. R) 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) 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 z. L) • Right Tail: β = (value from table based on z. R) • Two Tails: β = (value from table based on z. R) – (value from table based on z. L) 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 H 1: 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?