Confidence intervals and Hypothesis testing

Transcription

1 Confidence intervals and Hypothesis testing Confidence intervals offer a convenient way of testing hypothesis (all three forms). Procedure 1. Identify the parameter of interest.. Specify the significance level. 3. Depending on the hypothesis, construct the appropriate C.I. and apply the test 3.1 Two-sided: 100(1 /)% C.I. If postulated value is not within the C.I., reject H One-sided: Reject H 0 if the postulated value is greater or lesser than the bound in 100(1 )% C.I., for the upper- and lower-tailed test, respectively. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 13

2 One sample t-test for mean: Example Example: Training Method A A training method A is considered effective if the mean score is at least 75 on a test conducted pos-training. Null: Mean score for a trainees is at least µ 0 = 75 given, x A = 70, s A =3.366, n A = 10 Solution: H a : µ<µ 0, t =( x µ 0 )/(s/ p n)= Critical value: t 0.05 (9) = Confidence region: ( 1, ]. Does not include the postulated value. Reject H 0 at =0.05. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 14

3 C.I. for differences in means Assumption: Random sample, unknown variance, normal population Unequal variances: µ 1 µ 4 x 1 x ± t / (ñ 1 ) s s 1 n 1 + s n 3 5 ñ 1 = 6 4 (s 1/n 1 + s /n ) 7 5 (5) (s 1 /n 1) n 1 + (s /n ) +1 n +1 Equal variance: µ 1 µ " x 1 x ± t / ( )s p r 1n1 + 1n # = n 1 + n (6) Paired difference (dependent populations): µ 1 µ d ± t / (n 1)s D / p n d = 1 n nx i=1 d i ; d i = x 1,i x,i s D = 1 n 1 nx (d i d) i=1 (7) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 15

4 Two sample t-tests for mean: Unequal variances Example: Comparing yield data It is desired to test if two reactors A and B have the same yield. From data, n 1 = 50, ȳ 1 = 75.5, s 1 =1.4314; n = 50, ȳ = 7.47, s =.764 Sol: H a : µ 1 µ 6=0, t =6.9, ñ 1 = 73, Confidence region: (.169, 3.94). Reject H 0 at =0.05. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 16

5 Two sample t-tests for mean: Unequal variances Example: Comparing yield data It is desired to test if two reactors A and B have the same yield. From data, n 1 = 50, ȳ 1 = 75.5, s 1 =1.4314; n = 50, ȳ = 7.47, s =.764 Sol: H a : µ 1 µ 6=0, t =6.9, ñ 1 = 73, Confidence region: (.169, 3.94). Reject H 0 at =0.05. Q: Suppose we wish to test Yield(A) - Yield(B) >. Then, the result? Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 17

6 Paired test: Example Example: Test a weight-loss program Suppose we wish to test in a weight-loss program if weight before is different from weight after. From data, n = 0, D =8.5, sd = Sol: H a : 6= 0,C.I.:(6.833, ). Reject H 0 at =0.05. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 18

7 Paired test: Example Example: Test a weight-loss program Suppose we wish to test in a weight-loss program if weight before is different from weight after. From data, n = 0, D =8.5, sd = Sol: H a : 6= 0,C.I.:(6.833, ). Reject H 0 at =0.05. Q.: Suppose a two sample t-test is used inadvertently. Then,? Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 19

8 C.I. for variance Assumption: Gaussian population, random samples Two-sided: (n 1)s /,n 1 apple apple (n 1)s 1 /,n 1 (8) One-sided: Lower and upper confidence bounds (n 1)s,n 1 apple, apple (n 1)s 1,n 1 (9) Two-sided C.I. for ratio of variances: s 1 s f 1 / (n 1,n 1 1) apple 1 apple s 1 s f / (n 1,n 1 1) (10) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 0

9 Two sample test for variance: Example Example: Comparing variances of yields To test: The variability in yields of two processes are identical. From data, n 1 = 50, s A =.05; n = 50,s B =7.64. Solution: H 0 : A = B, H a : A 6= B, Statistic: f =0.7. Boundaries: f 0.05 (49, 49) = 0.567,f (49, 49) = C.I.: 1 [0.15, 0.473]. Reject H 0 at =0.05. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 1

10 C.I.s for proportion Two-sided: ˆp z / r ˆp(1 ˆp) n r ˆp(1 apple p apple ˆp + z / n ˆp) (11) One-sided: Left- and right-sided confidence bounds r ˆp(1 p apple ˆp + z n ˆp), ˆp z r ˆp(1 ˆp) n apple p (1) Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing

11 Tests for proportion: Example 1 Example: Defective controllers Manufacturer claims a maximum of 5% defective controllers. A random sample of n = 00 devices drawn reveals x =4(four) defective items. If the customer wishes to test that the proportion of defective items exceeds p 0 =0.05, H a : p>p 0 with H 0 : p = p 0. Solution: I 0 =(0.004, 0.096). Therefore,samplelargeenough. Statistic: z = Critical value: z c = C.I.: p [0.0037, 1). Fail to reject H 0 at =0.05. Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 3

12 Bibliography I Montgomery, D. C. and G. C. Runger (011). Applied Statistics and Probability for Engineers. 5 th edition. New York, USA: John Wiley & Sons, Inc. Ogunnaike, B. A. (010). Random Phenomena: Fundamentals of Probability and Statistics for Engineers. Raton, FL, USA: CRC Press, Taylor & Francis Group. Boca Arun K. Tangirala, IIT Madras Intro to Statistical Hypothesis Testing 4