Document Sample

Point Estimation and Confidence Intervals Estimation of Population Parameters • Statistical inference refers to making inferences about a population parameter through the use of sample information • The sample statistics summarize sample information and can be used to make inferences about the population parameters • Two approaches to estimate population parameters – Point estimation: Obtain a value estimate for the population parameter – Interval estimation: Construct an interval within which the population parameter will lie with a certain probability Point Estimation • In attempting to obtain point estimates of population parameters, the following questions arise – What is a point estimate of the population mean? – How good of an estimate do we obtain through the methodology that we follow? • Example: What is a point estimate of the average yield on ten- year Treasury bonds? • To answer this question, we use a formula that takes sample information and produces a number Point Estimation • A formula that uses sample information to produce an estimate of a population parameter is called an estimator • A specific value of an estimator obtained from information of a specific sample is called an estimate • Example: We said that the sample mean is a good estimate of the population mean – The sample mean is an estimator – A particular value of the sample mean is an estimate Point Estimation • Note: An estimator is a random variable that takes many possible values (estimates) • Question: Is there a unique estimator for a population parameter? For example, is there only one estimator for the population mean? • The answer is that there may be many possible estimators • Those estimators must be ranked in terms of some desirable properties that they should exhibit Properties of Point Estimators • The choice of point estimator is based on the following criteria – Unbiasedness – Efficiency – Consistency • A point estimator ˆ is said to be an unbiased estimator of the population parameter if its expected value (the mean of its sampling distribution) is equal to the population parameter it is trying to estimate ˆ E Properties of Point Estimators • Interesting Results on Unbiased Estimators – The sample mean, variance and proportion are unbiased estimators of the corresponding population parameters – Generally speaking, the sample standard deviation is not an unbiased estimator of the population standard deviation • We can also define the bias of an estimator as follows ˆ ˆ Bias E Properties of Point Estimators • It is usually the case that, if there is an unbiased estimator of a population parameter, there exist several others, as well • To select the “best unbiased” estimator, we use the criterion of efficiency • An unbiased estimator is efficient if no other unbiased estimator of the particular population parameter has a lower sampling distribution variance Properties of Point Estimators • If ˆ1 and ˆ2 are two unbiased estimators of the population parameter , then ˆ1 is more efficient than ˆ2 if ˆ ˆ V 1 V 2 • The unbiased estimator of a population parameter with the lowest variance out of all unbiased estimators is called the most efficient or minimum variance unbiased estimator • In some cases, we may be interested in the properties of an estimator in large samples, which may not be present in the case of small samples Properties of Point Estimators • We say that an estimator is consistent if the probability of obtaining estimates close to the population parameter increases as the sample size increases • The problem of selecting the most appropriate estimator for a population parameter is quite complicated • In some occasions, we may prefer to have some bias of the estimator at the gain of increases efficiency Properties of Point Estimators • One measure of the expected closeness of an estimator to the population parameter is its mean squared error MSE E ˆ ˆ 2 Interval Estimation • Point estimates of population parameters are prone to sampling error and are not likely to equal the population parameter in any given sample • Moreover, it is often the case that we are interested not in a point estimate, but in a range within which the population parameter will lie • An interval estimator for a population parameter is a formula that determines, based on sample information, a range within which the population parameter lies with certain probability Interval Estimation • The estimate is called an interval estimate • The probability that the population parameter will lie within a confidence interval is called the level of confidence and is given by 1 - • Two interpretations of confidence intervals – Probabilistic interpretation – Practical interpretation Interval Estimation • In the probabilistic interpretation, we say that – A 95% confidence interval for a population parameter means that, in repeated sampling, 95% of such confidence intervals will include the population parameter • In the practical interpretation, we say that – We are 95% confident that the 95% confidence interval will include the population parameter Constructing Confidence Intervals • Confidence intervals have similar structures Point Estimate Reliability Factor Standard Error – Reliability factor is a number based on the assumed distribution of the point estimate and the level of confidence – Standard error of the sample statistic providing the point estimate Confidence Interval for Mean of a Normal Distribution with Known Variance • Suppose we take a random sample from a normal distribution with unknown mean, but known variance • We are interested in obtaining a confidence interval such that it will contain the population mean 90% of times • The sample mean will follow a normal distribution and the corresponding standardized variable will follow a standard normal distribution Confidence Interval for Mean of a Normal Distribution with Known Variance • If X is the sample mean, then we are interested in the confidence interval, such that the following probability is .9 .9 P 1.645 Z 1.645 X P 1.645 1.645 / n 1.645 1.645 P X n n 1.645 1.645 P X X n n Confidence Interval for Mean of a Normal Distribution with Known Variance • Following the above expression for the structure of a confidence interval, we rewrite the confidence interval as follows X 1.645 n • Note that from the standard normal density PZ 1.65 FZ 1.65 0.95 P( Z 1.65) FZ 1.65 0.05 Confidence Interval for Mean of a Normal Distribution with Known Variance • Given this result and that the level of confidence for this interval (1-) is .90, we conclude that – The area under the standard normal to the left of –1.65 is 0.05 – The area under the standard normal to the right of 1.65 is 0.05 • Thus, the two reliability factors represent the cutoffs -z/2 and z/2 for the standard normal Confidence Interval for Mean of a Normal Distribution with Known Variance • In general, a 100(1-)% confidence interval for the population mean when we draw samples from a normal distribution with known variance 2 is given by X z / 2 n where z/2 is the number for which PZ z / 2 2 Confidence Interval for Mean of a Normal Distribution with Known Variance • Note: We typically use the following reliability factors when constructing confidence intervals based on the standard normal distribution – 90% interval: z0.05 = 1.65 – 95% interval: z0.025 = 1.96 – 99% interval: z0.005 = 2.58 • Implication: As the degree of confidence increases the interval becomes wider Confidence Interval for Mean of a Normal Distribution with Known Variance • Example: Suppose we draw a sample of 100 observations of returns on the Nikkei index, assumed to be normally distributed, with sample mean 4% and standard deviation 6% • What is the 95% confidence interval for the population mean? • The standard error is .06/ 100 = .006 • The confidence interval is .04 1.96(.006) Confidence Interval for Mean of a Normal Distribution with Unknown Variance • In a more typical scenario, the population variance is unknown • Note that, if the sample size is large, the previous results can be modified as follows – The population distribution need not be normal – The population variance need not be known – The sample standard deviation will be a sufficiently good estimator of the population standard deviation • Thus, the confidence interval for the population mean derived above can be used by substituting s for Confidence Interval for Mean of a Normal Distribution with Unknown Variance • However, if the sample size is small and the population variance is unknown, we cannot use the standard normal distribution • If we replace the unknown with the sample st. deviation s the following quantity X t s/ n follows Student’s t distribution with (n – 1) degrees of freedom Confidence Interval for Mean of a Normal Distribution with Unknown Variance • The t-distribution has mean 0 and (n – 1) degrees of freedom • As degrees of freedom increase, the t-distribution approaches the standard normal distribution • Also, t-distributions have fatter tails, but as degrees of freedom increase (df = 8 or more) the tails become less fat and resemble that of a normal distribution Confidence Interval for Mean of a Normal Distribution with Unknown Variance • In general, a 100(1-)% confidence interval for the population mean when we draw small samples from a normal distribution with an unknown variance 2 is given by s X tn1, / 2 n where tn-1,/2 is the number for which Ptn 1 tn1, / 2 2 Confidence Interval for Mean of a Normal Distribution with Unknown Variance • Example: Suppose we want to estimate a 95% confidence interval for the average quarterly returns of all fixed-income funds in the US • We assume that those returns are normally distributed with an unknown variance • We draw a sample of 150 observations and calculate the sample mean to be .05 and the standard deviation .03 Confidence Interval for Mean of a Normal Distribution with Unknown Variance • To find the confidence interval, we need tn-1,/2 = t149,0.025 • From the tables of the t-distribution, this is equal to 1.96 • The confidence interval is .03 .05 1.96 150 Confidence Interval for the Population Variance of a Normal Population • Suppose we have obtained a random sample of n observations from a normal population with variance 2 and that the sample variance is s2. A 100(1 - )% confidence interval for the population variance is n 1s 2 2 n 1s 2 2 2 n 1, / 2 n 1,1 / 2 Confidence Interval for the Population Variance of a Normal Population • The values of the chi-squared distribution with n-1,/2 and n-1,1-/2 are determined as follows 2 2 P n1 n1, / 2 2 P n1 n1,1 / 2 2 2 2 2 where n 1 follows the chi-squared distribution with (n-1) degrees of freedom

DOCUMENT INFO

Shared By:

Tags:
Point Estimation and Confidence Intervals, point estimation, Confidence intervals, hypothesis testing, random variables, sampling distributions, confidence interval, probability and statistics, Analysis of Variance, Statistical Inference

Stats:

views: | 677 |

posted: | 4/28/2009 |

language: | English |

pages: | 30 |

OTHER DOCS BY juanagui

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.