Sampling Distribution of a Sample Mean Lecture 27 Section 8.4 Tue, Mar 4, 2008

Sampling Distribution of the Sample Mean

With or Without Replacement? If the sample size is small in relation to the population size (< 5%), then it does not matter whether we sample with or without replacement. The calculations are simpler if we sample with replacement. In any case, we are not going to worry about it.

Experiment – Tenure of Senators State Years AL 10 CT 26 IL 30 MD 20 NE 6 NC 2 SC 4 VA 28 DE 8 MA 45 ND 15 WA AK 39 34 IA 22 NV SD 14 5 FL MI OH WV 48 AZ 12 KS NH TN GA 11 MN OK 13 WI 18 AR KY NJ 1 TX HI 7 MS OR WY CA 44 LA NM 24 UT ID 16 MO PA CO ME NY VT IN MT RI 32 29

Experiment – Tenure of Senators State Years 1 10 14 26 27 30 40 20 53 6 66 2 79 4 92 28 15 8 41 45 54 67 80 93 3 39 16 34 29 22 42 55 68 81 94 5 17 43 56 69 82 95 48 12 18 31 44 57 70 83 96 19 32 11 58 71 13 84 97 7 33 46 59 72 85 98 21 47 60 73 86 99 9 35 61 24 74 87 100 23 36 49 62 75 88 37 50 63 76 89 25 38 51 64 77 90 52 65 78 91

The Central Limit Theorem Begin with a population that has mean  and standard deviation . For sample size n, the sampling distribution of the sample mean is approximately normal if n  30, with

The Central Limit Theorem The approximation gets better and better as the sample size gets larger and larger. That is, the sampling distribution “morphs” from the original distribution to the normal distribution.

The Central Limit Theorem For many populations, the distribution is almost exactly normal when n  10. For almost all populations, if n  30, then the distribution is almost exactly normal.

The Central Limit Theorem Also, if the original population is exactly normal, then the sampling distribution of the sample mean is exactly normal for any sample size. This is all summarized on pages 536 – 537.