MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal Distributions 6 4 Sampling Distributions and Estimators 6 5 The Central Limit Theorem 6 6 Normal as Approximation to Binomial 6 7 Assessing Normality Key Concept The main objective of this section is to understand the concept of a sampling distribution of a statistic, which is the distribution of all values of that statistic when all possible samples of the same size are taken from the same population. We will also see that some statistics are better than others for estimating population parameters. Check out the Sample Means (xls) Excel program in the Technology section, at http://cfcc.edu/faculty/cmoore/indexexchtm.htm, of the Important Links webpage or go directly to http://cfcc.edu/faculty/cmoore/samplemeans.xls Properties Definitions The sampling distribution of a statistic (such as the sample mean or sample proportion) is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. (The sampling distribution of a statistic is typically represented as a probability distribution in the format of a table, probability histogram, or formula.) The sampling distribution of the mean is the distribution of sample means, with all samples having the same sample size n taken from the same population. (The sampling distribution of the mean is typically represented as a probability distribution in the format of a table, probability histogram, or formula.) Sample means target the value of the population mean. (That is, the mean of the sample means is the population mean. The expected value of the sample mean is equal to the population mean.) The distribution of the sample means tends to be a normal distribution. 1
Definition The sampling distribution of the variance is the distribution of sample variances, with all samples having the same sample size n taken from the same population. (The sampling distribution of the variance is typically represented as a probability distribution in the format of a table, probability histogram, or formula.) Properties Sample variances target the value of the population variance. (That is, the mean of the sample variances is the population variance. The expected value of the sample variance is equal to the population variance.) The distribution of the sample variances tends to be a distribution skewed to the right. Definition The sampling distribution of the proportion is the distribution of sample proportions, with all samples having the same sample size n taken from the same population. Definition We need to distinguish between a population proportion p and some sample proportion: p = population proportion = sample proportion Properties Sample proportions target the value of the population proportion. (That is, the mean of the sample proportions is the population proportion. The expected value of the sample proportion is equal to the population proportion.) The distribution of the sample proportion tends to be a normal distribution. Unbiased Sample means, variances, and proportions are unbiased estimators. That is they target the population parameter. These statistics are better in estimating the population parameter. Biased Sample medians, ranges, and standard deviations are biased estimators. That is they do NOT target the population parameter. Note: the bias with the standard deviation is relatively small in large samples so s is often used to estimate σ. 2
Specific results from 10,000 trials Consider repeating this process: Roll a die 5 times, find the mean, variance s 2, and the proportion of odd numbers of the results. What do we know about the behavior of all sample means that are generated as this process continues indefinitely? All outcomes are equally likely so the population mean is 3.5; the mean of the 10,000 trials is 3.49. If continued indefinitely, the sample mean will be 3.5. Also, notice the distribution is normal. Specific results from 10,000 trials Specific results from 10,000 trials All outcomes are equally likely so the population variance is 2.9; the mean of the 10,000 trials is 2.88. If continued indefinitely, the sample variance will be 2.9. Also, notice the distribution is skewed to the right. All outcomes are equally likely so the population proportion of odd numbers is 0.50; the proportion of the 10,000 trials is 0.50. If continued indefinitely, the mean of sample proportions will be 0.50. Also, notice the distribution is approximately normal. 3
Why Sample with Replacement? Sampling without replacement would have the very practical advantage of avoiding wasteful duplication whenever the same item is selected more than once. However, we are interested in sampling with replacement for these two reasons: 1. When selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement. 2. Sampling with replacement results in independent events that are unaffected by previous outcomes, and independent events are easier to analyze and result in simpler calculations and formulas. Caution Many methods of statistics require a simple random sample. Some samples, such as voluntary response samples or convenience samples, could easily result in very wrong results. Recap In this section we have discussed: Sampling distribution of a statistic. Sampling distribution of the mean. Sampling distribution of the variance. Sampling distribution of the proportion. Estimators. 292/10. In Exercises 9 12, refer to the population and list of samples in Example 4. Sampling Distribution of the Standard Deviation. From Example 4: Sampling Distribution of the Range Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 10 (based on Data Set 22 in Appendix B). Consider the values of 2, 3, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and 10. 292/12. In Exercises 9 12, refer to the population and list of samples in Example 4. Sampling Distribution of the Mean. From Example 4: Sampling Distribution of the Mean Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 10 (based on Data Set 22 in Appendix B). Consider the values of 2, 3, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and 10. 4
292/14. Assassinated Presidents: Sampling Distribution of the Median The ages (years) of the four U. S. presidents when they were assassinated in office are 56 (Lincoln), 49 (Garfield), 58 (McKinley), and 46 (Kennedy). a. Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. b. Find the median of each of the 16 samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution. (Use a format similar to Table 6 5 on page 289). c. Compare the population median to the mean of the sample medians. d. Do the sample medians target the value of the population median? In general, do sample medians make good estimators of population median? Why or why not? 292/14. 292/16. Assassinated Presidents: Sampling Distribution of the Variance The ages (years) of the four U. S. presidents when they were assassinated in office are 56 (Lincoln), 49 (Garfield), 58 (McKinley), and 46 (Kennedy). a. Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. b. Find the variance of each of the 16 samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution. (Use a format similar to Table 6 5 on page 289). c. Compare the population variance to the mean of the sample variances. d. Do the sample variances target the value of the population variance? In general, do sample variances make good estimators of population variance? Why or why not? 5
292/18. Births: Sampling Distribution of Proportion When 3 births are randomly selected, the sample space is bbb, bbg, bgb, bgg, gbb, gbg, ggb, and ggg. Assume that those 8 outcomes are equally likely. Describe the sampling distribution of the proportion of girls from 3 births as a probability distribution table. Does the mean of the sample proportions equal the proportion of girls in 3 births? (Hint: See Example 5.) 292/16. 292/21. Using a Formula to Describe a Sampling Distribution Example 5 includes a table and graph to describe the sampling distribution of the proportions of girls from 2 births. Consider the formula shown below, and evaluate that formula using sample proportions x of 0, 0.5, and 1. Based on the results, does the formula describe the sampling distribution? Why or why not? 292/20. Quality Control: Sampling Distribution of Proportion After constructing a new manufacturing machine, 5 prototype integrated circuit chips are produced and it is found that 2 are defective (D) and 3 are acceptable (A). Assume that two of the chips are randomly selected with replacement from this population. a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions of defects. b. Find the mean of the sampling distribution. c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of defects? Does the mean of the sampling distribution of proportions always equal the population proportion? Use TI calculator and let Y1 = P(x). 2ND Window, TblStart=0, ΔTbl=0.5 6