Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) α ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) α ✲ µ 0 Lower tail test for µ ≥ µ 0 : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) α ✲ µ 0 Lower 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) α 2 ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) α 2 ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) α α 2 2 ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) α α 2 2 ✲ µ 0 Two tailed test for µ � = µ 0 : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Consider a random sample from a normal population with known standard deviation σ . To test the null hypothesis µ = µ 0 against various alternative hypotheses, we use the test statistic z = x − µ 0 σ / √ n and define the following rejection regions. 1. For H a : µ > µ 0 use z ≥ z α ( upper tailed test ) 2. For H a : µ < µ 0 use z ≤ − z α ( lower tailed test ) 3. For H a : µ � = µ 0 use | z | ≥ z α 2 ( two-tailed test ) α α 2 2 ✲ µ 0 Two tailed 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 Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. 2. Determine the null hypothesis H 0 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. 2. Determine the null hypothesis H 0 . 3. Determine the alternative hypothesis H a . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. 2. Determine the null hypothesis H 0 . 3. Determine the alternative hypothesis H a . 4. Choose the appropriate test statistic. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. 2. Determine the null hypothesis H 0 . 3. Determine the alternative hypothesis H a . 4. Choose the appropriate test statistic. 5. Determine the rejection region using the significance level. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. 2. Determine the null hypothesis H 0 . 3. Determine the alternative hypothesis H a . 4. Choose the appropriate test statistic. 5. Determine the rejection region using the significance level. 6. Determine if the test statistic falls into the rejection region or not. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Performing Hypothesis Tests 1. Determine the parameter of interest. 2. Determine the null hypothesis H 0 . 3. Determine the alternative hypothesis H a . 4. Choose the appropriate test statistic. 5. Determine the rejection region using the significance level. 6. Determine if the test statistic falls into the rejection region or not. Reject or don’t reject accordingly. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Example. A new type of body armor is tested if it satisfies the specification of at most µ 0 = 1 . 9 in of displacement when hit with a certain type of bullet. The manufacturer tests by firing one round each at 36 samples of the new armor and measuring the displacement upon impact. The result is a sample mean displacement of 1 . 91 in . Assume the displacements are normally distributed with mean µ and a standard deviation of 0 . 06 in . Test if the armor is up to specifications at the 10 % significance level. Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z = x − µ 0 σ / √ n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z = x − µ 0 σ / √ n = 1 . 91 − 1 . 9 √ 0 . 06 / 36 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z = x − µ 0 σ / √ n = 1 . 91 − 1 . 9 = 0 . 01 √ 0 . 01 0 . 06 / 36 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z = x − µ 0 σ / √ n = 1 . 91 − 1 . 9 = 0 . 01 √ 0 . 01 = 1 0 . 06 / 36 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z = x − µ 0 σ / √ n = 1 . 91 − 1 . 9 = 0 . 01 √ 0 . 01 = 1 0 . 06 / 36 Decide if to reject or not. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Parameter of interest: µ , the true mean displacement. Null hypothesis: H 0 : µ = 1 . 9 in (null value of µ 0 ) Alternative hypothesis: H a : µ > 1 . 9 in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x − µ 0 σ / √ n = x − 1 . 9 0 . 06 / √ n Rejection region: We use an upper tailed test and the rejection region is z > z 0 . 1 ≈ 1 . 28. Substitute values into test statistic: z = x − µ 0 σ / √ n = 1 . 91 − 1 . 9 = 0 . 01 √ 0 . 01 = 1 0 . 06 / 36 Decide if to reject or not. Do not reject the null hypothesis, because the test statistic is not in the rejection region. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion A type II error is more serious than a type I error here. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. But when a test says “reject”, you really reject. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. But when a test says “reject”, you really reject. That’s it. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Discussion A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. But when a test says “reject”, you really reject. That’s it. Even if it’s expensive. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . β ( µ ′ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . X < µ 0 + z α σ / √ n | µ = µ ′ � β ( µ ′ ) � = P logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . X < µ 0 + z α σ / √ n | µ = µ ′ � β ( µ ′ ) � = P � X − µ ′ σ / √ n < z α + µ 0 − µ ′ � = σ / √ n P logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . X < µ 0 + z α σ / √ n | µ = µ ′ � β ( µ ′ ) � = P � X − µ ′ σ / √ n < z α + µ 0 − µ ′ � = σ / √ n P z α + µ 0 − µ ′ � � σ / √ n = Φ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . X < µ 0 + z α σ / √ n | µ = µ ′ � β ( µ ′ ) � = P � X − µ ′ σ / √ n < z α + µ 0 − µ ′ � = σ / √ n P z α + µ 0 − µ ′ � � σ / √ n = Φ ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . X < µ 0 + z α σ / √ n | µ = µ ′ � β ( µ ′ ) � = P � X − µ ′ σ / √ n < z α + µ 0 − µ ′ � = σ / √ n P z α + µ 0 − µ ′ � � σ / √ n = Φ ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
Testing Normal Distributions Example Sample Size Determination t -tests Probability of a type II error for H a : µ > µ 0 . X < µ 0 + z α σ / √ n | µ = µ ′ � β ( µ ′ ) � = P � X − µ ′ σ / √ n < z α + µ 0 − µ ′ � = σ / √ n P z α + µ 0 − µ ′ � � σ / √ n = Φ α ✲ µ 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means
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