Hypothesis Tests for Population Proportions Bernd Schr oder logo1 - PowerPoint PPT Presentation

Large Sample Size Small Sample Size Hypothesis Tests for Population Proportions Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) α ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) α ✲ µ 0 Upper tail test for µ ≤ µ 0 : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) α ✲ µ 0 Upper tail test for µ ≤ µ 0 : Tail probability is ≤ α (small) if µ ≤ µ 0 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ◮ For H a : p < p 0 use z ≤ − z α (lower tailed test) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ◮ For H a : p < p 0 use z ≤ − z α (lower tailed test) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ◮ For H a : p < p 0 use z ≤ − z α (lower tailed test) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ◮ For H a : p < p 0 use z ≤ − z α (lower tailed test) ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

Large Sample Size Small Sample Size To test H 0 : p = p 0 ... ... as long as np 0 ≥ 10 and n ( 1 − p 0 ) ≥ 10. (So that the normal approximation for the binomial distribution works.) p − p 0 ˆ 1. Test statistic: z = � p 0 ( 1 − p 0 ) / n 2. Alternative hypotheses and rejection regions. ◮ For H a : p > p 0 use z ≥ z α (upper tailed test) ◮ For H a : p < p 0 use z ≤ − z α (lower tailed test) ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions