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Calculation and Behavior of Type I and Type II Errors

AP Statistics. Calculation and Behavior of Type I and Type II Errors. Mr. Killian February, 2005. Definitions of Error Types. Type I error – Concluding that the null Hypothesis ( H o ) is false when in reality it is true. Probability of Type I error = 

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Calculation and Behavior of Type I and Type II Errors

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  1. AP Statistics Calculation and Behavior of Type I and Type II Errors Mr. Killian February, 2005

  2. Definitions of Error Types • Type I error – Concluding that the null Hypothesis (Ho) is false when in reality it is true. Probability of Type I error =  • Type II error – Concluding that the null hypothesis is true when in reality it is false. Probability of Type II error =  • Power = 1 -  : The probability of correctly concluding that the null hypothesis is false

  3. Calculating Type II Error Probability • To calculate the probability of a Type II Error, two things are needed • A hypothesis test, with specified Ho, Ha, , n, o, and  • A specific alternative value of the population mean, a

  4. Sampling Distribution if Ho is true Ho:  = o Ha: >o Sig. Lev.  Rejection Region Acceptance Region  o Steps to Calculating  1. Set up Hypothesis Test to define Rejection Region and Acceptance Region

  5. Sampling Distribution if Ho is true Rejection Region Acceptance Region  o Sampling Distribution if  = a a Steps to Calculating  2. Draw the sampling distribution for  = a

  6. Sampling Distribution if Ho is true Rejection Region Acceptance Region  o Sampling Distribution if  = a a  Steps to Calculating  3. The area of the distribution with  = athat overlaps with the acceptance region is

  7. Example • Mr. Postman Makes a Strict Budget of $400 for his Gas Bill. He is concerned that his budgeting is off and that this will damage his finances: • Ho: = $400 • Ha: >$400 •  = $50 • n = 24 •  = 0.1 • Find  when  = $425.

  8. Calculations • Standard deviation of sampling distribution • z-score for rejection region •  = 0.1, 1-tail: z* = 1.28 • Value of sample mean to reject • . • Reject for sample mean > $413.1

  9. Example: Calculating  Sampling Distribution if Ho is true  o=$400 Sampling Distribution if  = $425 • =normalcdf(-1E99, 413.1, 425, 10.2) =0.1217  a=$425

  10. o=$400  What happens if?  is reduced while parameters and n stay the same  o= $400  a=$425 a=$425 BEFORE:  = 0.1 AFTER : = 0.02  increases and Power decreases

  11. What happens if? a moves further from o while everything else is unchanged   o=$400 o=$400   a=$425 a=$450 BEFORE: a=$425 AFTER: a=$450 decreases and Power increases

  12. What happens if? n increases while all parameters are unchanged   o=$400 o=$400   a=$425 a=$425 BEFORE: n = 24 AFTER: n = 48  and decrease and Power increases

  13. Summary • A Type II error means that the null hypothesis is accepted when it is false • Calculation of a Type II error depends on a specific, given alternative value of the population mean:  = a •  and  demonstrate the following behavior: • If  is lowered,  increases (and power decreases) • If  is raised,  decreases (and power increases) • If a is changed to a value further from o, then  decreases (and power increases), and vice versa • If n increases  and  decrease (and power increases), and vice versa

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