advertisement

Sampling Distributions of Sample Mean Philadelphia Rainfall Data Population: 540 months of rainfall in Philadelphia (100ths of inches) from January, 1825-December, 1869. Histogram with superimposed Normal curve: 70 60 50 40 30 20 Std. Dev = 191.77 10 Mean = 367.7 0 0. 0 N = 540.00 10 20 30 40 50 60 70 80 90 10 11 12 13 14 15 16 0 0 0 0 0 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 00. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 RA IN Descriptive Statistics: Descriptive Statistics N RAIN 540 Valid N (listwise) 540 Minimum 19.00 Maximum 1582.00 Mean 367.6796 Std. Deviation 191.77305 Samples: Took 1000 random samples of size 30. Theoretical Mean and Standard Deviation (Standard Error) of Distribution of Sample Means: Mean : Y 367.68 Std. Dev. : Y n 191.77 30 35.0 Shape :Approximat ely Normal Histogram of 1000 sample means (n=30) with superimposed normal curve 140 120 100 80 60 40 Std. Dev = 33.58 20 Mean = 366.4 N = 1000.00 0 0 0. 46 0 0. 44 0 0. 42 0 0. 40 0 0. 38 0 0. 36 0 0. 34 0 0. 32 0 0. 30 0 0. 28 0 0. 26 Y BAR Descriptive Statistics: Descriptive Statistics N 1000 YBAR Valid N (listwise) Minimum 259.10 Maximum 469.30 Mean 366.4056 Std. Deviation 33.58160 1000 Note: The mean of the sample means (366.41) is very close to the population mean (367.68). However, the standard deviation of the sample means (33.58) is further away, specifically below the theoretical standard deviation of the sampling distribution (35.0). This is (partly) due to the fact that we are sampling from a finite population, and that our sample size is relatively large as compared to the population size. When these sizes are known, we can incorporate the finite population correction factor for the standard deviation (standard error) of the sampling distribution of the sample mean: Y n N n 191.77 540 30 35.0(.9727) 34.1 N 1 540 1 30 Note that in practice, either the population size is unknown or very large (even infinite), relative to the sample size, and this correction is ignored.