Chapter 9 Understanding Sampling Distributions Statistics as Random

9. 1 Parameters, Statistics, and Statistical Inference A statistic is a numerical value computed from a sample. Its value may differ for different samples. e. g. sample mean , sample standard deviation s, and sample proportion. A parameter is a numerical value associated with a population. Considered fixed and unchanging. e. g. population mean m, population standard deviation s, and population proportion p. Copyright © 2011 Brooks/Cole, Cengage Learning 2

Statistical Inference: making conclusions about population parameters on basis of sample statistics. Two most common procedures: Confidence intervals: an interval of values that the researcher is fairly sure will cover the true, unknown value of the population parameter. Hypothesis tests: uses sample data to attempt to reject a hypothesis about the population. Copyright © 2011 Brooks/Cole, Cengage Learning 3

9. 2 From Curiosity to Questions about Parameters The Big Five Parameters Copyright ©

Paired Differences Paired data (or paired samples): when pairs of variables are collected. Only interested in population (and sample) of differences, and not in the original data. • Each person measured twice. Two measurements of same characteristic or trait are made under different conditions. • Similar individuals are paired prior to an experiment. Each member of a pair receives a different treatment. Same response variable is measured for all individuals. • Two different variables are measured for each individual. Interested in amount of difference between two variables. Copyright © 2011 Brooks/Cole, Cengage Learning 5

Independent Samples Two samples are called independent samples when the measurements in one sample are not related to the measurements in the other sample. • Random samples taken separately from two populations and same response variable is recorded. • One random sample taken and a variable recorded, but units are categorized to form two populations. • Participants randomly assigned to one of two treatment conditions, and same response variable is recorded. Copyright © 2011 Brooks/Cole, Cengage Learning 6

Familiar Examples Translated into Questions about Parameters Situation 1. Estimating the proportion falling into a category of a categorical variable. Example research questions: What proportion of American adults believe there is extraterrestrial life? In what proportion of British marriages is the wife taller than her husband? Population parameter: p = proportion in the population falling into that category. Sample estimate: = proportion in the sample falling into that category. Copyright © 2011 Brooks/Cole, Cengage Learning 7

Familiar Examples Situation 2. Estimating the difference between two populations with regard to the proportion falling into a category of a qualitative variable. Example research questions: How much difference is there between the proportions that would quit smoking if taking the antidepressant buproprion (Zyban) versus if wearing a nicotine patch? How much difference is there between men who snore and men who don’t snore with regard to the proportion who have heart disease? Population parameter: p 1 – p 2 = difference between the two population proportions. Sample estimate: = difference between the two sample proportions. Copyright © 2011 Brooks/Cole, Cengage Learning 8

Familiar Examples Situation 3. Estimating the mean of a quantitative variable. Example research questions: What is the mean time that college students watch TV per day? What is the mean pulse rate of women? Population parameter: m = population mean for the variable Sample estimate: = sample mean for the variable Copyright © 2011 Brooks/Cole, Cengage Learning 9

Familiar Examples Situation 4. Estimating the mean of paired differences for quantitative variables. Example research questions: What is the mean difference in weights for freshmen at the beginning and end of the first semester? What is the mean difference in age between husbands and wives in Britain? Population parameter: m d = population mean of differences Sample estimate: = mean of differences for paired sample Copyright © 2011 Brooks/Cole, Cengage Learning 10

Familiar Examples Situation 5. Estimating the difference between two populations with regard to the mean of a quantitative variable. Example research questions: How much difference is there in average weight loss for those who diet compared to those who exercise to lose weight? How much difference is there between the mean foot lengths of men and women? Population parameter: m 1 – m 2 = difference between the two population means. Sample estimate: = difference between the two sample means. Copyright © 2011 Brooks/Cole, Cengage Learning 11

9. 3 SD Mod 0: An Overview of Sampling Distribution Statistics as Random Variables Each new sample taken value of the sample statistic will change. The distribution of possible values of a statistic for repeated samples of the same size from a population is called the sampling distribution of the statistic. Many statistics of interest have sampling distributions that are approximately normal distributions Copyright © 2011 Brooks/Cole, Cengage Learning 12

Example 9. 2 Mean Hours of Sleep for College Students Survey of n = 190 college students. “How many hours of sleep did you get last night? ” Sample mean = 7. 1 hours. If we repeatedly took samples of 190 and each time computed the sample mean, the histogram of the resulting sample mean values would look like the histogram at the right: Copyright © 2011 Brooks/Cole, Cengage Learning 13

Standard Deviation and Standard Error of a Statistic • Standard deviation of a sampling distribution measures the variation among possible values of the sample statistic over all possible random samples. We include the name of the statistic being studied, e. g. the standard deviation of the mean. • Standard error describes the estimated value of the standard deviation of a statistic. We include the name of the statistic, e. g. the standard error of the mean. Copyright © 2011 Brooks/Cole, Cengage Learning 14

9. 4 SD Mod 1: Sampling Distribution for One Sample Proportion • Suppose (unknown to us) 40% of a population carry the gene for a disease, (p = 0. 40). • We will take a random sample of 25 people from this population and count X = number with gene. • Although we expect (on average) to find 10 people (40%) with the gene, we know the number will vary for different samples of n = 25. • In this case, X is a binomial random variable with n = 25 and p = 0. 4. Copyright © 2011 Brooks/Cole, Cengage Learning 15

Many Possible Samples Four possible random samples of 25 people: Sample 1: X =12, proportion with gene =12/25 = 0. 48 or 48%. Sample 2: X = 9, proportion with gene = 9/25 = 0. 36 or 36%. Sample 3: X = 10, proportion with gene = 10/25 = 0. 40 or 40%. Sample 4: X = 7, proportion with gene = 7/25 = 0. 28 or 28%. Note: • Each sample gave a different answer, which did not always match the population value of 40%. • Although we cannot determine whether one sample will accurately reflect the population, statisticians have determined what to expect for most possible samples. Copyright © 2011 Brooks/Cole, Cengage Learning 16

Sampling Distribution for a Sample Proportion Let p = population proportion of interest or binomial probability of success. Let = sample proportion or proportion of successes. If numerous random samples or repetitions of the same size n are taken, the distribution of possible values of is approximately a normal curve distribution with • Mean = p • Standard deviation = s. d. ( ) = This approximate distribution is sampling distribution of Copyright © 2011 Brooks/Cole, Cengage Learning . 17

The Normal Curve Approximation Rule for Sample Proportions Normal Approximation Rule can be applied in two situations: Situation 1: A random sample is taken from a population. Situation 2: A binomial experiment is repeated numerous times. In each situation, three conditions must be met: 1: The Physical Situation There is an actual population or repeatable situation. 2: Data Collection A random sample is obtained or situation repeated many times. 3: The Size of the Sample or Number of Trials The size of the sample or number of repetitions is relatively large, np and np(1 -p) must be at least 5 and preferable at least 10. Copyright © 2011 Brooks/Cole, Cengage Learning 18

Examples for which Rule Applies • Election Polls: to estimate proportion who favor a candidate; units = all voters. • Television Ratings: to estimate proportion of households watching TV program; units = all households with TV. • Consumer Preferences: to estimate proportion of consumers who prefer new recipe compared with old; units = all consumers. • Testing ESP: to estimate probability a person can successfully guess which of 5 symbols on a hidden card; repeatable situation = a guess. Copyright © 2011 Brooks/Cole, Cengage Learning 19

Example 9. 4 Possible Sample Proportions Favoring a Candidate Suppose 40% all voters favor Candidate C. Pollsters take a sample of n = 2400 voters. Rule states the sample proportion who favor X will have approximately a normal distribution with mean = p = 0. 4 and s. d. ( ) = Histogram at right shows sample proportions resulting from simulating this situation 400 times. Copyright © 2011 Brooks/Cole, Cengage Learning 20

Estimating the Population Proportion from a Single Sample Proportion In practice, we don’t know the true population proportion p, so we cannot compute the standard deviation of , s. d. ( )= . In practice, we only take one random sample, so we only have one sample proportion. Replacing p with in the standard deviation expression gives us an estimate that is called the standard error of. s. e. ( )= . If = 0. 39 and n = 2400, then the standard error is 0. 01. So the true proportion who support the candidate is almost surely between 0. 39 – 3(0. 01) = 0. 36 and 0. 39 + 3(0. 01) = 0. 42. Copyright © 2011 Brooks/Cole, Cengage Learning 21

9. 5 SD Mod 2: Sampling Distribution for Diff in Two Sample Proportions For the populations: p 1 = population proportion for the first population. p 2 = population proportion for the second population. Parameter: p 1 – p 2 = difference in popul proportions. For the samples: = sample proportion for sample from first popul. = sample proportion for sample from second popul. Statistic: = difference in sample proportions. Copyright © 2011 Brooks/Cole, Cengage Learning 22

Conditions Sampling distribution of difference in two independent sample proportions is approximately normal when: Condition 1: Sample proportions are available for two independent samples, randomly selected from the two populations of interest. Condition 2: All of the quantities n 1 p 1, n 1(1 – p 1), n 2 p 2, and n 1(1 – p 2) are at least 10. These quantities represent the expected numbers of successes and failures in each sample. Copyright © 2011 Brooks/Cole, Cengage Learning 23

Sampling Distribution for the Difference in Two Sample Proportions Mean = p 1 – p 2 Standard deviation = s. d. ( ) = When we don’t know the populations proportions, we use the sample proportions, resulting in: Standard error = s. e. ( Copyright © 2011 Brooks/Cole, Cengage Learning )= 24

Example 9. 6 Men, Women, Death Penalty Suppose 37% of women and 27% of men oppose death penalty, p 1 =. 37 and p 2 =. 27, for a difference p 1 – p 2 =. 37 –. 27 =. 10 For independent random samples of 1017 women and 885 men, the sampling distribution of is approx normal with mean. 10 and standard deviation: Note: 2008 survey gave observed difference of. 36 –. 285 =. 075, which is not unusual. Copyright © 2011 Brooks/Cole, Cengage Learning 25

9. 6 SD Mod 3: Sampling Distribution for One Sample Mean • Suppose we want to estimate the mean weight loss for all who attend clinic for 10 weeks. Suppose (unknown to us) the distribution of weight loss is approximately N(8 pounds, 5 pounds). • We will take a random sample of 25 people from this population and record for each X = weight loss. • We know the value of the sample mean will vary for different samples of n = 25. • What do we expect those means to be? Copyright © 2011 Brooks/Cole, Cengage Learning 26

Many Possible Samples Four possible random samples of 25 people: Sample 1: Mean = 8. 32 pounds, standard deviation = 4. 74 pounds. Sample 2: Mean = 6. 76 pounds, standard deviation = 4. 73 pounds. Sample 3: Mean = 8. 48 pounds, standard deviation = 5. 27 pounds. Sample 4: Mean = 7. 16 pounds, standard deviation = 5. 93 pounds. Note: • Each sample gave a different answer, which did not always match the population mean of 8 pounds. • Although we cannot determine whether one sample mean will accurately reflect the population mean, statisticians have determined what to expect for most possible sample means. Copyright © 2011 Brooks/Cole, Cengage Learning 27

The Normal Curve Approximation Rule for Sample Means Let m = mean for population of interest. Let s = standard deviation for population of interest. Let = sample mean. If numerous random samples of the same size n are taken, the distribution of possible values of is approximately a normal curve distribution with • Mean = m • Standard deviation = s. d. ( ) = This approximate distribution is sampling distribution of Copyright © 2011 Brooks/Cole, Cengage Learning . 28

The Normal Curve Approximation Rule for Sample Means Normal Approximation Rule can be applied in two situations: Situation 1: The population of measurements of interest is bell-shaped and a random sample of any size is measured. Situation 2: The population of measurements of interest is not bell-shaped but a large random sample is measured. Note: Difficult to get a Random Sample? Researchers usually willing to use Rule as long as they have a representative sample with no obvious sources of confounding or bias. Copyright © 2011 Brooks/Cole, Cengage Learning 29

Examples for which Rule Applies • Average Weight Loss: to estimate average weight loss; weight assumed bell-shaped; population = all current and potential clients. • Average At Death: to estimate average at which left-handed adults (over 50) die; ages at death not bell-shaped so need n 30; population = all left-handed people who live to be at least 50. • Average Student Income: to estimate mean monthly income of students at university who work; incomes not bell-shaped and outliers likely, so need large random sample of students; population = all students at university who work. Copyright © 2011 Brooks/Cole, Cengage Learning 30

Example 9. 8 Hypothetical Mean Weight Loss Suppose the distribution of weight loss is approximately N(8 pounds, 5 pounds) and we will take a random sample of n = 25 clients. Rule states the sample mean weight loss will have a normal distribution with mean = m = 8 pounds and s. d. ( ) = pound Histogram at right shows sample means resulting from simulating this situation 400 times. Empirical Rule: It is almost certain that the sample mean will be between 5 and 11 pounds. Copyright © 2011 Brooks/Cole, Cengage Learning 31

Standard Error of the Mean In practice, the population standard deviation s is rarely known, so we cannot compute the standard deviation of , s. d. ( )= . In practice, we only take one random sample, so we only have the sample mean and the sample standard deviation s. Replacing s with s in the standard deviation expression gives us an estimate that is called the standard error of. s. e. ( )= . For a sample of n = 25 weight losses, the standard deviation is s = 4. 74 pounds. So the standard error of the mean is 0. 948 pounds. Copyright © 2011 Brooks/Cole, Cengage Learning 32

Increasing the Size of the Sample Suppose we take n = 100 people instead of just 25. The standard deviation of the mean would be s. d. ( ) = pounds. • For samples of n = 25, sample means are likely to range between 8 ± 3 pounds => 5 to 11 pounds. • For samples of n = 100, sample means are likely to range only between 8 ± 1. 5 pounds => 6. 5 to 9. 5 pounds. Larger samples tend to result in more accurate estimates of population values than smaller samples. Copyright © 2011 Brooks/Cole, Cengage Learning 33

9. 7 SD Mod 4: Sampling Distribution for Sample Mean of Paired Differences Let md = mean for population of differences. Let sd = standard deviation for population of differences. Let = mean for the sample of differences. Let sd = standard deviation for sample of differences. If numerous random samples of the same size n are taken, the distribution of possible values of is approximately a normal curve distribution with • Mean = md • Standard deviation = s. d. ( ) = This approximate distribution is sampling distribution of Copyright © 2011 Brooks/Cole, Cengage Learning . 34

The Normal Curve Approximation Rule for Sample Mean of Paired Differences Normal Approximation Rule can be applied in two situations: Situation 1: The population of differences is bell-shaped and a random sample of any size is measured. Situation 2: The population of differences is not bell-shaped but a large random sample is measured. Note: Difficult to get a Random Sample? Researchers usually willing to use Rule as long as they have a representative sample with no obvious sources of confounding or bias. Copyright © 2011 Brooks/Cole, Cengage Learning 35

Standard Error of the Mean Difference Standard deviation of : s. d. ( Standard error of )= . : s. e. ( ) = . The standard error is used to estimate the standard deviation. Copyright © 2011 Brooks/Cole, Cengage Learning 36

Example 9. 9 No “Freshman 15” How likely to see a mean weight gain of 4. 2 pounds or larger (for a random sample of 60 freshman) if there is no average weight gain in the population of all such students? Suppose the standard deviation for the population of weight gains is known to be 7 pounds. The sampling distribution of is … • Approximately normal • mean = md = 0 pounds • s. d. ( ) = pounds Empirical Rule: 95% of the possible sample means will be between -1. 8 and 1. 8 pounds. Copyright © 2011 Brooks/Cole, Cengage Learning 37

9. 8 SD Mod 5: Sampling Distribution for Difference in Two Sample Means Let m 1 = population mean for first population. Let m 2 = population mean for second population. Parameter: m 1 – m 2 = difference in population means. Let = sample mean for sample from first population. Let = sample mean for sample from second population. Statistic: = difference in sample means. Let s 1 = population standard deviation for first population. Let s 2 = population standard deviation for second population. Let s 1 = sample std deviation for sample from first population. Let s 2 = sample std deviation for sample from second population. Copyright © 2011 Brooks/Cole, Cengage Learning 38

Conditions for Sampling Distribution of to be Approx Normal An important condition in this situation is that the two samples must be independent. How? • Take separate random samples from each of two populations such as men and women. • Take a random sample from a population and divide the sample into two groups based on a categorical variable such as smoker and nonsmoker. • Randomly assign participants in a randomized experiment to two treatment groups such as exercise or diet. Copyright © 2011 Brooks/Cole, Cengage Learning 39

Conditions for Sampling Distribution of to be Approx Normal In addition to independent samples, one of the following two situations must hold: Situation 1: The populations of measurements are both bell-shaped and random samples of any size are measured. Situation 2: Large random samples are measured from each population. Arbitrary definition of large is both samples are at least 30, but extreme outliers or extreme skewness in either sample may require even larger samples. Copyright © 2011 Brooks/Cole, Cengage Learning 40

Standard Error of the Mean Difference Standard deviation of s. d. ( Standard error of s. e. ( : )= . The standard error is used to estimate the standard deviation. Copyright © 2011 Brooks/Cole, Cengage Learning 41

Example 9. 10 Who Are the Speed Demons? What’s the fastest you’ve ever driven a car? ____ mph. Mean for 87 males = 107 mph, mean for 102 females = 88 mph. Is this 19 mph difference large enough to convince of real difference in populations? Suppose standard deviations for each population of speeds is known to be 15 mph. The sampling distribution of is: • Approximately normal • mean = m 1 – m 2 = 0 mph • s. d. ( )= Note: difference of 19 mph almost impossible in this scenario. Thus, true difference in population means almost surely much greater than 0. Copyright © 2011 Brooks/Cole, Cengage Learning 42

9. 9 Preparing for Statistical Inference: Standardized Statistics If conditions are met, these standardized statistics have, approximately, a standard normal distribution N(0, 1). Copyright © 2011 Brooks/Cole, Cengage Learning 43

Example 9. 11 Unpopular TV Shows Networks cancel shows with low ratings. Ratings based on random sample of households, using the sample proportion watching show as estimate of population proportion p. If p < 0. 20, show will be cancelled. Suppose in a random sample of 1600 households, 288 are watching (for proportion of 288/1600 = 0. 18). Is it likely to see = 0. 18 even if p were 0. 20 (or higher)? The sample proportion of 0. 18 is about 2 standard deviations below the mean of 0. 20. Copyright © 2011 Brooks/Cole, Cengage Learning 44

Student’s t-Distribution: Replacing s with s Dilemma: we generally don’t know s. Using s we have: If sample size n is small, this standardized statistic will not have a N(0, 1) distribution but rather a t-distribution with n – 1 degrees of freedom (df). More on t-distributions in Chapters 11 and 13. Copyright © 2011 Brooks/Cole, Cengage Learning 45

Example 9. 12 Standardized Mean Weights Claim: mean weight loss is m = 8 pounds. Sample of n =25 people gave a sample mean weight loss of = 8. 32 pounds and a sample standard deviation of s = 4. 74 pounds. Is the sample mean of 8. 32 pounds reasonable to expect if m = 8 pounds? The sample mean of 8. 32 is only about one-third of a standard error above 8, which is consistent with a population mean weight loss of 8 pounds. Copyright © 2011 Brooks/Cole, Cengage Learning 46