Chapter 8 Estimation

Estimator and Estimate An estimator of a population parameter is a random variable that depends on the sample information and whose value provides approximations to this unknown parameter. A specific value of that random variable is called an estimate.

Point Estimator and Point Estimate Let  represent a population parameter (such as the population mean  or the population proportion ). A point estimator, , of a population parameter, , is a function of the sample information that yields a single number called a point estimate. For example, the sample mean, , is a point estimator of the population mean , and the value that assumes for a given set of data is called the point estimate.

Unbiasedness The point estimator is said to be an unbiased estimator of the parameter  if the expected value, or mean, of the sampling distribution of is ; that is,

Probability Density Functions for unbiased and Biased Estimators (Figure 8.1)

Bias Let be an estimator of . The bias in is defined as the difference between its mean and ; that is It follows that the bias of an unbiased estimator is 0.

Most Efficient Estimator and Relative Efficiency Suppose there are several unbiased estimators of . Then the unbiased estimator with the smallest variance is said to be the most efficient estimator or to be the minimum variance unbiased estimator of . Let and be two unbiased estimators of , based on the same number of sample observations. Then, a) is said to be more efficient than if b) The relative efficiency of with respect to is the ratio of their variances; that is,

Point Estimators of Selected Population Parameters (Table 8.1) Properties Mean,  X Unbiased, Most Efficient (assuming normality) Xm Unbiased (assuming normality), but not most efficient Proportion,  p Unbiased, Most Efficient Variance, 2 s2

Confidence Interval Estimator A confidence interval estimator for a population parameter  is a rule for determining (based on sample information) a range, or interval that is likely to include the parameter. The corresponding estimate is called a confidence interval estimate.

Confidence Interval and Confidence Level Let  be an unknown parameter. Suppose that on the basis of sample information, random variables A and B are found such that P(A <  < B) = 1 - , where  is any number between 0 and 1. If specific sample values of A and B are a and b, then the interval from a to b is called a 100(1 - )% confidence interval of . The quantity of (1 - ) is called the confidence level of the interval. If the population were repeatedly sampled a very large number of times, the true value of the parameter  would be contained in 100(1 - )% of intervals calculated this way. The confidence interval calculated in this manner is written as a <  < b with 100(1 - )% confidence.

P(-1.96 < Z < 1.96) = 0.95, where Z is a Standard Normal Variable (Figure 8.3) 0.025 0.025 -1.96 1.96

Notation Let Z/2 be the number for which where the random variable Z follows a standard normal distribution.

Selected Values Z/2 from the Standard Normal Distribution Table (Table 8.2) 0.01 0.02 0.05 0.10 Z/2 2.58 2.33 1.96 1.645 Confidence Level 99% 98% 95% 90%

Confidence Intervals for the Mean of a Population that is Normally Distributed: Population Variance Known Consider a random sample of n observations from a normal distribution with mean  and variance 2. If the sample mean is X, then a 100(1 - )% confidence interval for the population mean with known variance is given by or equivalently, where the margin of error (also called the sampling error, the bound, or the interval half width) is given by

Basic Terminology for Confidence Interval for a Population Mean with Known Population Variance (Table 8.3) Terms Symbol To Obtain: Standard Error of the Mean Z Value (also called the Reliability Factor) Use Standard Normal Distribution Table Margin of Error Lower Confidence Limit Upper Confidence Limit Width (width is twice the bound)

Student’s t Distribution Given a random sample of n observations, with mean X and standard deviation s, from a normally distributed population with mean , the variable t follows the Student’s t distribution with (n - 1) degrees of freedom and is given by

Notation A random variable having the Student’s t distribution with v degrees of freedom will be denoted tv. The tv,/2 is defined as the number for which

Confidence Intervals for the Mean of a Normal Population: Population Variance Unknown Suppose there is a random sample of n observations from a normal distribution with mean  and unknown variance. If the sample mean and standard deviation are, respectively, X and s, then a 100(1 - )% confidence interval for the population mean, variance unknown, is given by or equivalently, where the margin of error, the sampling error, or bound, B, is given by and tn-1,/2 is the number for which The random variable tn-1 has a Student’s t distribution with v=(n-1) degrees of freedom.

Confidence Intervals for Population Proportion (Large Samples) Let p denote the observed proportion of “successes” in a random sample of n observations from a population with a proportion  of successes. Then, if n is large enough that (n)()(1- )>9, then a 100(1 - )% confidence interval for the population proportion is given by or equivalently, where the margin of error, the sampling error, or bound, B, is given by and Z/2, is the number for which a standard normal variable Z satisfies

Notation A random variable having the chi-square distribution with v = n-1 degrees of freedom will be denoted by 2v or simply 2n-1. Define as 2n-1, the number for which

The Chi-Square Distribution (Figure 8.17) 1 -   2n-1,

The Chi-Square Distribution for n – 1 and (1-)% Confidence Level (Figure 8.18) /2 /2 1 -  2n-1,1- /2 2n-1,/2

Confidence Intervals for the Variance of a Normal Population Suppose there is a random sample of n observations from a normally distributed population with variance 2. If the observed variance is s2 , then a 100(1 - )% confidence interval for the population variance is given by where 2n-1,/2 is the number for which and 2n-1,1 - /2 is the number for which And the random variable 2n-1 follows a chi-square distribution with (n – 1) degrees of freedom.

Confidence Intervals for Two Means: Matched Pairs Suppose that there is a random sample of n matched pairs of observations from a normal distributions with means X and Y . That is, x1, x2, . . ., xn denotes the values of the observations from the population with mean X ; and y1, y2, . . ., yn the matched sampled values from the population with mean Y . Let d and sd denote the observed sample mean and standard deviation for the n differences di = xi – yi . If the population distribution of the differences is assumed to be normal, then a 100(1 - )% confidence interval for the difference between means (d = X - Y) is given by or equivalently,

Confidence Intervals for Two Means: Matched Pairs (continued) Where the margin of error, the sampling error or the bound, B, is given by And tn-1,/2 is the number for which The random variable tn – 1, has a Student’s t distribution with (n – 1) degrees of freedom.

Confidence Intervals for Difference Between Means: Independent Samples (Normal Distributions and Known Population Variances) Suppose that there are two independent random samples of nx and ny observations from normally distributed populations with means X and Y and variances 2x and 2y . If the observed sample means are X and Y, then a 100(1 - )% confidence interval for (X - Y) is given by or equivalently, where the margin of error is given by

Confidence Intervals for Two Means: Unknown Population Variances that are Assumed to be Equal Suppose that there are two independent random samples with nx and ny observations from normally distributed populations with means X and Y and a common, but unknown population variance. If the observed sample means are X and Y, and the observed sample variances are s2X and s2Y, then a 100(1 - )% confidence interval for (X - Y) is given by or equivalently, where the margin of error is given by

Confidence Intervals for Two Means: Unknown Population Variances that are Assumed to be Equal (continued) The pooled sample variance, s2p, is given by is the number for which The random variable, T, is approximately a Student’s t distribution with nX + nY –2 degrees of freedom and T is given by,

Confidence Intervals for Two Means: Unknown Population Variances, Assumed Not Equal Suppose that there are two independent random samples of nx and ny observations from normally distributed populations with means X and Y and it is assumed that the population variances are not equal. If the observed sample means and variances are X, Y, and s2X , s2Y, then a 100(1 - )% confidence interval for (X - Y) is given by where the margin of error is given by

Confidence Intervals for Two Means: Unknown Population Variances, Assumed Not Equal (continued) The degrees of freedom, v, is given by If the sample sizes are equal, then the degrees of freedom reduces to

Confidence Intervals for the Difference Between Two Population Proportions (Large Samples) Let pX, denote the observed proportion of successes in a random sample of nX observations from a population with proportion X successes, and let pY denote the proportion of successes observed in an independent random sample from a population with proportion Y successes. Then, if the sample sizes are large (generally at least forty observations in each sample), a 100(1 - )% confidence interval for the difference between population proportions (X - Y) is given by Where the margin of error is

Sample Size for the Mean of a Normally Distributed Population with Known Population Variance Suppose that a random sample from a normally distributed population with known variance 2 is selected. Then a 100(1 - )% confidence interval for the population mean extends a distance B (sometimes called the bound, sampling error, or the margin of error) on each side of the sample mean, if the sample size, n, is

Sample Size for Population Proportion Suppose that a random sample is selected from a population. Then a 100(1 - )% confidence interval for the population proportion, extending a distance of at most B on each side of the sample proportion, can be guaranteed if the sample size, n, is

Key Words Bias Bound Confidence interval: Confidence Level Estimate For mean, known variance For mean, unknown variance For proportion For two means, matched For two means, variances equal For two means, variances not equal For variance Confidence Level Estimate Estimator Interval Half Width Lower Confidence Limit (LCL) Margin of Error Minimum Variance Unbiased Estimator Most Efficient Estimator Point Estimate Point Estimator

Key Words (continued) Relative Efficiency Reliability Factor Sample Size for Mean, Known Variance Sample Size for Proportion Sampling Error Student’s t Unbiased Estimator Upper Confidence Limit (UCL) Width