Confidence intervals: Calculating z* Example 1: What is the z* for a C' = 95% confidence interval? Standard Normal Distribition hobobley 3 - N(0, 1) Todie way for 2 is the area under to 1 = 0 dandard naval curve 95% 0 =1 TABLE A Standard normal probabilities TAME A Standard normal probabilkies fromvingred 14 9279 5919 -3.4 D.I SITE 3517 3475 5714 5753 3.1 -3.0 10013 AM3 0013 0012 AME Doll -20 015 5015 0014 DOE -28 1 21 2.5% 2.5% .7517 21 0 8134 -26 -25 052 1.0 8577 .1509 8621 24 2.5% STU 8729 6 013 0120 4125 6 0113 0110 97.5% 13 WHY SIGE SITT -21 21 = -1.96 9365 22 = 1.96 14 SHIN -19 1244 1439 MORE 8525 4535 4583 1,7 0359 4351 10344 0836 4179 -IT 0455 8436 0437 0418 4409 0401 0502 4394 0173 0367 95% within 0 1 1.96 x 1 1.9 5750 53512 ANIT Therefore, z* = 1.96 Notice that 1.96 ~ 2, as the 68-95-99.7 rule suggestsQuestion 9: (2 points) Data was collected and a 99% confidence interval for #, the population mean, was found to be (99.164, 104.316). Assuming that the population was normally distributed with o = 10, would you reject Ho : / = 100 vs Ha : # # 100 at a = 0.05? (a) Yes (b) No (c) Unlikely to (d) Insufficient information Answer 9: (b) From the 99% confidence interval given we can find the sample mean (mid-point) and the margin of error (one-half of the interval width). So a = (104.316 + 99.164)/2 = 101.74 and z*o/ vn = 104.316 -101.74) = 2.576, where z* is the positive z-score for the 99% confidence interval (that is = 2.33). Therefore, o/ Vn = 1.106. Next we consider the hypothesis test using the same sample. First compute the z-statistic for the test: zstat = (X - HO)/(o/ vn) = (101.74-100)/1.106 = 1.573, corresponding to a p-value of 2 x 5.82% (it's a two-sided test!). Because the p-value is larger than o, we fail to reject the null hypothesis