Two Sample Hypothesis Tests

Transcription

1 Note Packet #21 Two Sample Hypothesis Tests CEE 3710 November 13, 2017 Review Possible states of nature: H o and H a (Null vs. Alternative Hypothesis) Possible decisions: accept or reject Ho (rejecting Ho implies accept Ha) Probability decide incorrectly: Type I error α = P[ Reject Ho Ho true ] Type II error β = P[ Accept Ho Ho false ] **Same concepts apply to Two Sample Hypothesis Tests 1

2 Two Sample Hypothesis Tests X ~ N[µ X, σ X2 ] {X i } are independent, i= 1,...,n x ~ N[µ, σ 2 ] { j } are independent, j = 1,...,n X and are independent Interested in µ X µ = E[X] E[] Test for difference in means H o : µ X = µ versus H a : µ X µ (Two Tailed Test) H a : µ X < µ (Lower Tail Test) H a : µ X > µ (Upper Tail Test) Examples: Are fish in stream X bigger than fish in stream? Compare the life of two different highway paving compounds. Test Statistics for Two Sample Hypothesis Tests For a one sample hypothesis test on μ, we define a test statistic which represents the standardized value of the sample mean under the null hypothesis H o : μ = μ o. For a single sample of size n drawn from a population characterized by the variable X ~ i.i.d. N[μ, σ 2 ], we consider test statistics of the form: X 2 o n or X 2 S n o 2

3 These statistics extend to two sample hypothesis tests on the difference in means (µ X µ ), wherein E[ X ] = µx - µ Var[ X ] = n X X for X n Case I: σ X and σ KNOWN (any n X, n ) Case II: σ X and σ UNKNOWN, but n X and n LARGE Case III: σ X and σ UNKNOWN, and n X and n SMALL Case I: σ X and σ known Given H o, the test statistic Z test (X ) ( X ) X n n X has a standard normal distribution. Use standard normal test. 3

4 Case II: σ X and σ unknown, but n X and n LARGE Given H o, the test statistic Z test (X ) ( X ) SX S n n X is nearly standard normal when we n X and n are large, such that the error in estimating σ X and σ by S X and S can be ignored. Use standard normal test. Case III: σ X and σ unknown, small samples Cannot ignore error in S x and S. Use: (X ) ( ) T S n X X S X n Given H o, the test statistic T has approximately Student t distribution with degrees of freedom: 2 sx s nx n X X nx 1 n 1 s /n s /n 4

5 NOTE: It is best not to use the Pooled t test discussed in Ayyub & McCuen (Eqns ) which assumes that σ X = σ, which may not be true. Fisheries Example A fisheries biologist is concerned with the average weights of salmon in two streams. There is a pollution problem in Stream, and the biologist is concerned that this is inhibiting the growth of the salmon. The data obtained for each stream is given below. Do fish in Stream X tend to be bigger than fish in Stream? Stream X Stream n X = 100 n = 140 s X = 0.32 s = 0.45 x = y =

6 Fisheries Example Do fish in Stream X tend to be bigger than fish in Stream? Stream X Stream n X = 100 n = 140 s X = 0.32 s = 0.45 x = y = (1) What are the appropriate hypotheses? (2) What is the appropriate test statistic? (3) What do you conclude at the 5% level? Groundwater Example An engineer is concerned that the lining of a landfill may be failing. To test if there is a problem, he measures nitrate concentrations at two wells: one upstream of the landfill and one downstream. Data for the two locations are provided below. Based on this data, is there leachate from the landfill? In other words, are nitrate concentrations greater downstream than upstream? Upstream Downstream x = 4.2 y = 5.6 s X = 2.1 s = 1.8 n X = 11 n = 6 6

7 Groundwater Example Are nitrate concentrations greater downstream than upstream? Upstream Downstream x = 4.2 y = 5.6 s X = 2.1 s = 1.8 n X = 11 n = 6 (1) What are the appropriate hypotheses? (2) What is the appropriate test statistic? (3) What is the rejection region for a test at the 5% level? (4) What do you conclude at the 5% level? 7

8 Stopping Distance Example Consider two different types of braking systems for cars: System X vs. System. Let µ X and µ represent the true mean stopping distances from 50 mph for cars of a certain type equipped with System X and System, respectively. It is believed that cars equipped with System X are able to stop over a much shorter distance than cars equipped with System. The following data were obtained for each breaking system: Upstream Downstream n X = 6 n = 6 x = ft y = ft s X = 5.03 ft s = 5.38 ft Stopping Distance Example The following data were obtained for each breaking system: Upstream Downstream n X = 6 n = 6 x = ft y = ft s X = 5.03 ft s = 5.38 ft State the appropriate hypotheses to test if the cars equipped with System X can stop 10 ft sooner than cars equipped with System. Use the data to test this hypothesis at the 1% level. What do you conclude? 8