36226 HW 9 solutions - Introduction to Statistical...

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Introduction to Statistical Inference 36-226 Assignment #9 – Solutions Spring, 2017 Problem #1 (a) H 0 : π = 0 . 50 vs H a : π = 0 . 50 . α = . 01 level test Reject H 0 if | z * | > z α/ 2 = z . 005 = 2 . 575 Data: n = 1003 x = 462 ˆ π = 462 1003 = . 461 Test Statistics: z * 1 = . 461 - . 50 ( . 461)(1 - . 461) 1003 = - . 039 . 01574 = - 2 . 478 or z * 2 = . 461 - . 50 ( . 50)(1 - . 50) 1003 = - . 039 . 01579 = - 2 . 470 Decision: Since | z * 1 = - 2 . 478 | < 2 . 575 or ( | z * 2 = - 2 . 470 | < 2 . 575) , we fail to reject H 0 : π = . 50 at the α = . 01 level. At α = . 01 level, we can not conclude that the proportion of cell phone owners who would use their phone to call for advice about a purchase is different from 50%. (b) p -value = 2 · P ( Z < z * 1 = - 2 . 478) = 2 · ( . 0068) = 0 . 0136 (or if you used z * 2 , p-value = .0132). (c) 99% large sample confidence interval for π : ˆ π ± z . 005 ˆ π (1 - ˆ π ) n . 461 ± 2 . 575 ( . 461)(1 - . 461) 1003 = . 461 ± (2 . 575)( . 01574) [ . 421 , . 502] We are 99% confident that the true proportion of cell phone owners who would use their phone to call for advice about a purchase is between 42.1% and 50.2%. Problem #2 The test statistics for testing H 0 : μ = 475 versus H a : μ > 475 is z * = ¯ x - 475 100 / n . We will use the p -value that corresponds to each test to determine whether there is sufficient evidence to reject H 0 . (a) z * = 478 - 475 100 / 100 = 0 . 3 , p -value = 0.38. Do not reject H 0 . (b) z * = 478 - 475 100 / 1000 = 0 . 95 , p -value = 0.17. Do not reject H 0 . (c) z * = 478 - 475 100 / 10000 = 3 , p -value = 0.001. Reject H 0 .
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