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The Probability of a Type II Error and the Power of the Test

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.

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The Probability of a Type II Error and the Power of the Test

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  1. The Probability of a Type II Error and the Power of the Test

  2. 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.

  3. Review: How to Determine Picture H1: p ≠ value Two Tails H1: p < value Left Tail H1: p > value Right Tail

  4. 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)

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

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

  7. 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)

  8. 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)

  9. Finding Probability of a Type II Error / Power of Test (Proportion) (By Hand)

  10. Finding Probability of a Type II Error / Power of Test (Proportion) (By Hand)

  11. 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)

  12. 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?

  13. 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?

  14. 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)

  15. Finding Probability of a Type II Error / Power of Test (Mean) (TI-83/84)

  16. Finding Probability of a Type II Error / Power of Test (Mean) (TI-83/84)

  17. 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)

  18. 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)

  19. Finding Probability of a Type II Error / Power of Test (Mean) (By Hand)

  20. Finding Probability of a Type II Error / Power of Test (Mean) (By Hand)

  21. 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)

  22. 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?

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