Question

NASA is conducting an experiment to find out the fraction of people who black out at G forces greater than 6. In an earlier study, the population proportion was estimated to be 0.36. How large a sample would be required in order to estimate the fraction of people who black out at 6 or more Gs at the 99% cofidence level with an error at most 0.02? **PLEASE EXPLAIN HOW TO READ z-table**

Answer 1

Sample Size for 99% CI, E-002 , Þ = 0.36 Confidence level 99% and Desired Margin of Error, E-0.02 Minimum Sam ple size is given by: note: Margin of Error = (Length of C1)/2 Best estimate is p-0.36 Significance level - a 1- Confidence 1-0.99-0.01 The Critical Value 4/2-20.005-2.58 (closest ualue From z table) Zo.005 n=px(1-p) 2.58 0.36 x 0.64 3834.086400 3834.086 Since the minimum n has to be integer, we take the ceiling of above number and get n3835 Sample Size should be atleast n3835 Note that if we take exact z-value, we will get a different answer as the exact calculation will be 2.5758293035489 0.02 0.36 × 0.64 3821.70044 n This will round up to: n 3822 Also, some people prefer using critical value of 2.575, which gives answer 3819.24 + 3820 Find: zo.01/2 Zo/2 is the z-score such that the area to the right of Zo/2 1s a/2 (under the standard normal curve

0.995 Z0.005 In a z-table having area to the left of z, look for value closest to 0.995 inside the table, and find it along row named 2.5 and column named 0.08 Hence our answer is 2.58 Let's also see what we get from linear interpolation (0.9950) is not present in the z-table containing area to left of z, but we can use linear interpolation We note that nearby 0.9949 has z-score 2.57 and 0.9951 has z-score 2.58 Thus we can use "linear interpolation" to find out approximate z-score -2.58 0.9950-0.9951 0.9950 is 0.9949-0.9951 - 1/2 way from 0.9951 to 0.9949 So, the z-score for 0.9950 is approzimately 1/2 of the way from 2.58 to 2.57 Hence z-score-2.584 ((2.57)-(2.58)) * 1/2-2.5750 Bonus fact: Using TI-83/84 function invNorm(0.995) or Excel function: NormInv(O exact z-score of 0.995 is 2.5758293035 Note that different methods give different answers, the "correct" answer will depend on instructor. Use interpolation if you were taught to use that. Here, we will round to 2 digits Z0.01/2 2.58 or 2.575 or 2.576