Introduction to Statistical Hypothesis Testing
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1 Introduction to Statistical Hypothesis Testing Arun K. Tangirala Power of Hypothesis Tests Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 1
2 Learning objectives I Computing Pr(Type II error) and Power I Choice of sample size I Closing remarks (for the course) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 2
3 Opening remarks The goodness or success of a hypothesis tests, in general, I Proper statement of alternative hypothesis, H a. I True variability, i.e.,of the population(s), 2. I Sample size n I Choice of significance level. A measure of the strength of a hypothesis test is its power, which in turn is, dependent on the Type II error. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 3
4 Type II error: One-sample test for mean The hypotheses are H 0 :µ = µ 0 H a :µ 6= µ 0 We fail to reject the null whenever the observed statistic Z o <z /2. Recall, for a two-tailed test for mean with known variance, Pr(Type II error) = Pr( Z o <z /2 H 0 is false) (1) Suppose that the null is false and that the true mean is µ 1 = µ 0 +. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 4
5 Computing Pr(Type II error): Two-tailed z-test for mean The test statistic is then Z o = X µ 0 / = X µ 1 / + (2) The distribution of the test-statistic is thus, Z o N, 1 (3) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 5
6 Computing probability of Type II error Introduce the standardized variable, Z 1 = X µ 1 / N(0, 1). Then, in terms of this newly introduced standardized variable, Z o = Z contd.. Therefore, from (1), we have = Pr Z 1 + <z /2 = Pr z /2 <Z 1 < z /2 (4) where the probability is computed using a standard Gaussian distribution. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 6
7 Computing probability of Type II error Introduce the standardized variable, Z 1 = X µ 1 / N(0, 1). Then, in terms of this newly introduced standardized variable, Z o = Z contd.. Therefore, from (1), we have = Pr Z 1 + <z /2 = Pr z /2 <Z 1 < z /2 (4) where the probability is computed using a standard Gaussian distribution. For a given, and sample size n, the power of hypothesis test is computed as Power =1 (5) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 7
8 Type II errors Two-tailed test: = F z /2 F z /2 (6) Lower-tailed test: =1 F z (7) Upper-tailed test: = F z (8) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 8
9 Example Propellant burning rate Consider the motivational and worked out example of the average propellant burning rate. Suppose that the true burning rate is µ = 49 cm/s. What is the Type II error for the two-sided test with =0.05, =2and sample size n = 25? Solution: The difference between postulated and truth is = 1 and z /2 =1.96. Therefore,from (4), we have = F z /2 F z /2 = F (4.46) F (0.54) = Thus, there is a 30% chance that the test will fail to reject H 0, i.e., the truth will be undetected. Observe that the answer would be the same for =1. The power of the test with this sample size is therefore 1 =0.7, which is satisfactory. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 9
10 Example Propellant burning rate Consider the motivational and worked out example of the average propellant burning rate. Suppose that the true burning rate is µ = 49 cm/s. What is the Type II error for the two-sided test with =0.05, =2and sample size n = 25? Solution: The difference between postulated and truth is = 1 and z /2 =1.96. Therefore,from (4), we have = F z /2 F z /2 = F (4.46) F (0.54) = Thus, there is a 30% chance that the test will fail to reject H 0, i.e., the truth will be undetected. Observe that the answer would be the same for =1. The power of the test with this sample size is therefore 1 =0.7, which is satisfactory. How can we improve the power of a hypothesis test? Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 10
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