LAB 7 FOR EXCEL 2007 SAMPLING DISTRIBUTIONS OF STATISTICS NAME__________________________LAB TIME_________PHYS ROOM NO____________ 1. 2. 3. Every statistic (such as the Sample Mean, the Sample Standard Deviation, the Sample Variance ) has a SAMPLING DISTRIBUTION which shows the pattern of variation of the statistic when the same population is sampled over and over and over. In this lab we will use Excel’s random number generator to draw 2500 observations from a population which has a Normal Distribution with a mean = 100 and a standard deviation = 5. What should be the approximate values of the smallest and largest of these 2500? Use + and – 3 sigma as the approximate “edge” of the normal distribution. Minimum should be approximately ( mean – 3 std dev) = ____________ 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. Maximum should be approximately ( mean + 3 std dev)= ____________ In Excel, do Data>Data Analysis>Random Number Generator. In the window which appears, set Variables = 5, Number of Random Numbers = 500, Distribution = Normal, Mean = 100, Standard Deviation = 5, Random Seed = 25674, click on Output Range, and enter A2 in the box beside the output range. Check that the random seed has not changed and then click OK. On line 1 of Excel, number the columns A through E as 1,2,3,4,5 We now have 500 lines of 5 observations each in Columns A through E. We will treat each line as a random sample of size = 5. So we have 500 random samples of size =5 in rows 2 through 501. CALCULATION OF MEAN, STANDARD DEVIATION AND RANGE FOR EACH SAMPLE: In G1 enter sample mean; in H1 enter sample std dev; in I1 enter sample max, in J1 enter sample min; in K1 enter sample range. In G2 calculate the mean for a2:e2. (Use the excel AVERAGE (a2:e2) In H2 calculate the standard deviation of a2:e2. Use excel STDEV (a2:e2) In I2 calculate the maximum for a2:e2. Use excel MAX (a2:e2) In J2 calculate the minimum for a2:e2. Use excel MIN (a2:e2) In K2 calculate the sample range, I2-J2. Copy G2:K2 down through line 501. In G503 calculate the overall mean of the 500 sample means. (Use excel AVERGE (g2:g501) Enter the overall mean of the 500 sample means _________________ Assuming that the POPULATION MEAN = 100, calculate the percent error for the mean of the 500 sample means using (G503-100)/100*100% :_________________ Does the sample mean appear to be an unbiased statistic?__________________ In G504 calculate the standard deviation of the 500 sample means. Use excel STDEV (g2:g501) 22. Enter the standard deviation of the 500 sample means:___________________ 23. Theory says the standard deviation of the sample means = Population std dev / sqrt (n). or 5 / sqrt(5) = 2.236. 24. Calculate the percent error for the value in step 22 vs the theoretical value in step 23 using (G504-2.236)/2.236*100% : ________________ 25. In H 503 calculate the mean of the 500 sample standard deviations: _________________ 26. Assuming that the POPULATION STANDARD DEVIATION = 5, calculate the percent error in the mean of the 500 sample standard deviations: ________________________ 27. Does the sample standard deviation appear to be an unbiased statistic? ______________ 28. Theory says that the sample standard deviation will average 6% lower than the population standard deviation when the sample size = 5. If this is true, the mean of the 500 sample standard deviations should have been 4.7000. Calculate the percent error between the actual mean of the 500 sample standard deviations (step 25) and the predicted value given in step 28: ____________________________ 29. 30. In K503 calculate the mean of the 500 sample ranges:__________________________ 31. Theory says that the sample range for a sample size = 5 will average 2.326 times the population standard deviation, or 5 (2.326) = 11.63. 32. Calculate the percent error for the actual mean of 500 sample ranges (step 30 ) vs the theoretical value (step 31): ____________________ 33. 34. 35. 36. 37. 38. 39. DRAWING A HISTOGRAM OF THE SAMPLE MEANS Do Tools>Data Analysis>Histogram and click OK. Input range: G2:G501. Click on Output Range, and in the box enter M1. Click on Chart Output, and click OK. Close up the spaces between the histogram bars. (right click on any bar, click on Format Data Series, click on Options and bring Gap down to 0. Size the histogram. Correct the title to read Histogram Of Sample Means. 40. Comment on the shape of the histogram: _____________________________________ 41. Print the histogram to turn in with these answers.