Sampling Distributions and Estimation Chapter 8 Chapter Contents 8.1 Sampling Variation 8.2 Estimators and Sampling Errors 8.3 Sample Mean and the Central Limit Theorem 8.4 Confidence Interval for a Mean (μ) with Known σ 8.5 Confidence Interval for a Mean (μ) with Unknown σ 8.6 Confidence Interval for a Proportion (π) 8.7 Estimating from Finite Populations 8.8 Sample Size Determination for a Mean 8.9 Sample Size Determination for a Proportion 8.10 Confidence Interval for a Population Variance,  2 (Optional)

Sampling Distributions and Estimation Chapter 8 Chapter Learning Objectives (LO’s) LO8-1: Define sampling error, parameter, and estimator. LO8-2: Explain the desirable properties of estimators. LO8-3: State the Central Limit Theorem for a mean. LO8-4: Explain how sample size affects the standard error. LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ.

Sampling Distributions and Estimation Chapter 8 Chapter Learning Objectives (LO’s) LO8-6: Know when to use Student’s t instead of z to estimate μ. LO8-7: Construct a 90, 95, or 99 percent confidence interval for π. LO8-8: Construct confidence intervals for finite populations. LO8-9: Calculate sample size to estimate a mean or proportion. LO8-10: Construct a confidence interval for a variance (optional).

8.1 Sampling Variation Chapter 8 Sample statistic – a random variable whose value depends on which items from the population happen to be included in the random sample. This is obviously dependent on random chance. Depending on the items selected, the sample statistic could either represent the population well or differ greatly from the population. If Sample size increases, then the sample statistic would represent the population parameter more faithfully. This sampling variation can easily be illustrated. 8-4

8.1 Sampling Variation Chapter 8 Consider eight random samples of size n = 5 from a large population of GMAT scores for MBA applicants. The sample means tend to be close to the population mean (parameter m = 520.78 ). 8-5

8.1 Sampling Variation Chapter 8 The dot plots show that the sample means have much less variation than the individual sample items. This is because the sample mean is an average.

Making Inferences About a Population Cannot control: Sampling variation Population variation Can Control: Sample size level of confidence that is needed in our estimate

Some Terminology 8.2 Estimators and Sampling Distributions Chapter 8 LO8-1: Define sampling error, parameter and estimator. Some Terminology Estimator – a statistic derived from a sample to infer the value of a population parameter. Estimate – the value of the estimator for a particular sample. Population parameters are usually represented by Greek letters and the corresponding statistic by Roman letters. 8-8

Examples of Estimators 8.2 Estimators and Sampling Distributions LO8-1 Chapter 8 Examples of Estimators Sampling Distributions Think of each estimator as a random variable. The sampling distribution of an estimator is the probability distribution of all possible values the statistic may assume when a random sample of size n is taken. Note: An estimator is a random variable since samples vary. It will have one value for each sample we take. In a real study we would take only one sample, but it helps to think of taking many samples when we are studying the behavior of estimators.

8.2 Estimators and Sampling Distributions LO8-1 Chapter 8 Sampling error is the difference between an estimate and the corresponding population parameter. For example, if we use the sample mean as an estimate for the population mean, then the Bias is the difference between the expected value of the estimator and the value of the true parameter. Example for the mean, An estimator is unbiased if its expected value is = the value of the parameter being estimated. The sample mean is an unbiased estimator of the population mean since On average, an unbiased estimator neither overstates nor understates the true parameter. 8-10

8.2 Estimators and Sampling Distributions LO8-1 Chapter 8 The dots indicate the different values of the estimators, one for each sample we take from the population. The bulls eye represents the single true value of the population parameter. 8-11

Efficiency 8.2 Estimators and Sampling Distributions Chapter 8 LO8-2 LO8-2: Explain the desirable properties of estimators. Note: Obviously, a desirable property for an estimator is for it to be unbiased. Efficiency Efficiency refers to the variance of the estimator’s sampling distribution. A more efficient estimator has smaller variance. If variance is smaller, then the value of the estimator is likely to be closer to the true value of the parameter, and we can use smaller sample sizes. Figure 8.6 What are MVUEs for population mean, population variance, population proportion? MVUE=minimum variance, unbiased estimator 8-12

8.2 Estimators and Sampling Distributions LO8-2 Chapter 8 LO8-2: Explain the desirable properties of estimators. Consistency A consistent estimator converges or gets closer in value toward the population parameter being estimated as the sample size increases. Figure 8.6 The PDFs are for sample means, for differing sample sizes. 8-13

Chapter 8 8.3 Sample Mean and the Central Limit Theorem LO8-3 LO8-3: State the Central Limit Theorem for a mean. The Central Limit Theorem is a powerful result that allows us to approximate the shape of the sampling distribution of the sample mean even when we don’t know what the population looks like, i.e., if it is normal or not. 8-14

Chapter 8 8.3 Sample Mean and the Central Limit Theorem LO8-3 As the sample size n increases, the distribution of sample means narrows in on the population mean µ. Standard error of the mean gets smaller. If the population is exactly normal, then the sample mean follows a normal distribution. 8-15

Chapter 8 8.3 Sample Mean and the Central Limit Theorem LO8-3 If the sample is large enough, the sample means will have approximately a normal distribution even if your population is not normal.

Illustrations of Central Limit Theorem 8.3 Sample Mean and the Central Limit Theorem LO8-3 Chapter 8 Illustrations of Central Limit Theorem Using the uniform and a right skewed distribution. Note: 8-17

Applying The Central Limit Theorem 8.3 Sample Mean and the Central Limit Theorem LO8-3 Chapter 8 Applying The Central Limit Theorem The Central Limit Theorem permits us to define an interval within which the sample means are expected to fall. As long as the sample size n is large enough, we can use the normal distribution regardless of the population shape (or any n if the population is normal to begin with). Discuss Example 8.1 in text, page 299. In class examples: 8.1, 8.3 8-18

Sample Size and Standard Error 8.3 Sample Mean and the Central Limit Theorem LO8-4 Chapter 8 LO8-4: Explain how sample size affects the standard error. Sample Size and Standard Error Even if the population standard deviation σ is large, the sample means will fall within a narrow interval as long as n is large. The key is the standard error of the mean:.. The standard error decreases as n increases. For example, when n = 4 the standard error is halved. To halve it again requires n = 16, and to halve it again requires n = 64. To halve the standard error, you must quadruple the sample size (the law of diminishing returns). 8-19

Illustration: All Possible Samples from a Uniform Population 8.3 Sample Mean and the Central Limit Theorem Chapter 8 Illustration: All Possible Samples from a Uniform Population Consider a discrete uniform population consisting of the integers {0, 1, 2, 3}. The population parameters are: m = 1.5, s = 1.118. 8-20

Illustration: All Possible Samples from a Uniform Population 8.3 Sample Mean and the Central Limit Theorem Chapter 8 Illustration: All Possible Samples from a Uniform Population The population is uniform, yet the distribution of all possible sample means of size 2 has a peaked triangular shape. 8-21

What is a Confidence Interval? 8.4 Confidence Interval for a Mean () with known () LO8-5 Chapter 8 LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ. What is a Confidence Interval? 8-22

What is a Confidence Interval? 8.4 Confidence Interval for a Mean () with known () LO8-5 Chapter 8 What is a Confidence Interval? The confidence interval for m with known s is: 8-23

Choosing a Confidence Level 8.4 Confidence Interval for a Mean () with known () LO8-5 Chapter 8 Choosing a Confidence Level A higher confidence level leads to a wider confidence interval. Greater confidence implies loss of precision (i.e. greater margin of error). 95% confidence is most often used. Confidence Intervals for Example 8.2 in Text 8-24

When Can We Assume Normality? 8.4 Confidence Interval for a Mean () with known () LO8-5 Chapter 8 Interpretation A confidence interval either does or does not contain m. The confidence level quantifies the risk. Out of 100 confidence intervals, approximately 95% may contain m, while approximately 5% might not contain  when constructing 95% confidence intervals. When Can We Assume Normality? If  is known and the population is normal, then we can safely use the formula to compute the confidence interval. If  is known and we do not know whether the population is normal, a common rule of thumb is that n  30 is sufficient to use the formula as long as the distribution is approximately symmetric with no outliers. Larger n may be needed to assume normality if you are sampling from a strongly skewed population or one with outliers. In class examples: 8.9, 8.13 8-25

Student’s t Distribution 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 LO8-6: Know when to use Student’s t instead of z to estimate . Student’s t Distribution Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation s is unknown and the sample size is small. 8-26

Student’s t Distribution 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 LO8-6: Know when to use Student’s t instead of z to estimate . Student’s t Distribution 8-27

Student’s t Distribution 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Student’s t Distribution Used instead of z when population is normal but population’s standard deviation is unknown. This is usually the case. t distributions are symmetric and shaped like the standard normal distribution. The t distribution is dependent on the size of the sample. Comparison of Normal and Student’s t Figure 8.11

Degrees of Freedom Chapter 8 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Degrees of Freedom Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation. The d.f for the t distribution in this case, is given by d.f. = n -1. This is because the sample mean is calculated as an intermediate step, thereby having only n-1 independent observations. As n increases, the t distribution approaches the shape of the normal distribution. For a given confidence level, t is always larger than z, so a confidence interval based on t is always wider than if z were used.

Comparison of z and t Chapter 8 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Comparison of z and t For very small samples, t-values differ substantially from the normal. As degrees of freedom increase, the t-values approach the normal z-values. For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 = 30. So for a 90 percent confidence interval, we would use t = 1.697, which is only slightly larger than z = 1.645.

Example GMAT Scores Again 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Example GMAT Scores Again Figure 8.13

Example GMAT Scores Again 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Example GMAT Scores Again Construct a 90% confidence interval for the mean GMAT score of all MBA applicants. x = 510 s = 73.77 (s is the sample standard deviation) Since s is unknown, use the Student’s t for the confidence interval with d.f. = 20 – 1 = 19. First find t/2 = t.05 = 1.729 from Appendix D.

Chapter 8 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 For a 90% confidence interval, use Appendix D to find t0.05 = 1.729 with d.f. = 19. Note: One can use Excel, Minitab, etc. to obtain these values as well as to construct confidence Intervals. We are 90 percent confident that the true mean GMAT score might be within the interval [481.48, 538.52]

Confidence Interval Width 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Confidence Interval Width Confidence interval width reflects - the sample size (greater the size, the smaller the interval), - the confidence level (more the confidence, the larger the interval) and - the standard deviation (more deviation means larger interval). To obtain a narrower interval and more precision - increase the sample size or - lower the confidence level (e.g., from 90% to 80% confidence). Outliers Outlier values that are beyond, say 3 standard deviations should be carefully examined. Are they real? They can skew the mean and sample variance. Usually, they are tossed out of the sample.

Using Appendix D Chapter 8 8.5 Confidence Interval for a Mean () with Unknown () LO8-6 Chapter 8 Using Appendix D Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10. If the table does not give the exact degrees of freedom, use the t-value for the next lower degrees of freedom. This is a conservative procedure since it causes the interval to be slightly wider. A conservative statistician may use the t distribution for confidence intervals when σ is unknown because using z would underestimate the margin of error. In class examples: 8.19, 8.21

Chapter 8 LO8-7 8.6 Confidence Interval for a Proportion () LO8-7: Construct a 90, 95, or 99 percent confidence interval for π. The distribution of a sample proportion p = x/n is symmetric if p = .50 and regardless of p, approaches symmetry as n increases. A proportion is a mean of data whose only values are 0 or 1. 8-36

LO8-7 8.6 Confidence Interval for a Proportion () Chapter 8 8-37

When is it Safe to Assume Normality of p? LO8-7 8.6 Confidence Interval for a Proportion () Chapter 8 When is it Safe to Assume Normality of p? Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both np 10 and n(1- p) 10. Sample size to assume normality: Table 8.9 8-38

Confidence Interval for p LO8-7 8.6 Confidence Interval for a Proportion () Chapter 8 Confidence Interval for p Since p is unknown, the confidence interval for p = x/n (assuming a large sample) is 8-39

Example Auditing Chapter 8 LO8-7 8.6 Confidence Interval for a Proportion () Chapter 8 Example Auditing In class Examples: 8.32 (a), 8.35 8-40

Chapter 8 LO8-8 8.7 Estimating from Finite Population LO8-8: Construct Confidence Intervals for Finite Populations. N = population size; n = sample size Should be unknown sigma

Sample Size to Estimate m LO8-9 8.8 Sample Size determination for a Mean Chapter 8 LO8-9: Calculate sample size to estimate a mean or proportion. Sample Size to Estimate m To estimate a population mean with a precision of + E (allowable error), you would need a sample of size n. Now,

How to Estimate s? We need it to calculate n: LO8-9 8.8 Sample Size determination for a Mean Chapter 8 How to Estimate s? We need it to calculate n: Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of s in the sample size formula. Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set s = [(b-a)/12]½. Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set s = (b-a)/4. This assumes normality with most of the data with m ± 2s so the range is 4s. Method 4: Poisson Arrivals In the special case when m is a Poisson arrival rate, then s = SQRT(Lambda). In class Examples: 8.43, 8.45 43

Chapter 8 8.9 Sample Size determination for a Proportion LO8-9 To estimate a population proportion with a precision of ± E (allowable error), you would need a sample of size Since p is a number between 0 and 1, the allowable error E is also between 0 and 1.

How to Estimate p? Chapter 8 8.9 Sample Size determination for a Proportion LO8-9 Chapter 8 How to Estimate p? Method 1: Assume that p = .50 This conservative method ensures the desired precision. However, the sample may end up being larger than necessary. Method 2: Take a Preliminary Sample Take a small preliminary sample and use the sample p in place of p in the sample size formula. Method 3: Use a Prior Sample or Historical Data How often are such samples available? Unfortunately, p might be different enough to make it a questionable assumption. In class Examples: 8.53 45

Homework 8.2 8.10 8.14 8.18 8.20 8.34 8.36 8.44