Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 Inferences Concerning Means

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 7.1 Point Estimation

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Point Estimator A point estimator of a parameter is a single number that can be regarded as a sensible value for. A point estimator can be obtained by selecting a suitable statistic and computing its value from the given sample data.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Unbiased Estimator A point estimator is said to be an unbiased estimator of if for every possible value of. If is not biased, the difference is called the bias of.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Bias of The pdf’s of a biased estimator and an unbiased estimator for a parameter

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Bias of The pdf’s of a biased estimator and an unbiased estimator for a parameter

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Some Unbiased Estimators When X is a binomial rv with parameters n and p, the sample proportion is an unbiased estimator of p. If X 1, X 2,…, X n is a random sample from a distribution with mean, then is an unbiased estimator of.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Some Unbiased Estimators Let X 1, X 2,…, X n be a random sample from a distribution with mean and variance. Then the estimator is an unbiased estimator.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Principle of Minimum Variance Unbiased Estimation (MVUE) Among all estimators of that are unbiased, choose the one that has the minimum variance. The resulting is called the minimum variance unbiased estimator (MVUE) of.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Graphs of the pdf’s of two different unbiased estimators

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. MVUE for a Normal Distribution Let X 1, X 2,…, X n be a random sample from a normal distribution with parameters and. Then the estimator is the MVUE for.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. A biased estimator that is preferable to the MVUE

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Error The standard error of an estimator is its standard deviation. If the standard error itself involves unknown parameters whose values can be estimated, substitution into yields the estimated standard error of the estimator, denoted

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 7.2 Interval Estimation

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Intervals An alternative to reporting a single value for the parameter being estimated is to calculate and report an entire interval of plausible values – a confidence interval (CI). A confidence level is a measure of the degree of reliability of the interval.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 95% Confidence Interval If after observing X 1 = x 1,…, X n = x n, we compute the observed sample mean, then a 95% confidence interval for the mean of normal population can be expressed if known as

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Other Levels of Confidence 0

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Other Levels of Confidence A confidence interval for the mean of a normal population when the value of is known is given by

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sample Size The general formula for the sample size n necessary to ensure an interval width w is

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large-Sample Confidence Interval Estimation for the Mean

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large-Sample Confidence Interval If n is sufficiently large, the standardized variable has approximately a standard normal distribution. This implies that is a large-sample confidence interval for with level

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large-Sample Confidence Bounds for Upper Confidence Bound: Lower Confidence Bound:

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Intervals Based on a Normal Population Distribution

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Normal Distribution The population of interest is normal, so that X 1,…, X n constitutes a random sample from a normal distribution with both

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. t -Distribution When is the mean of a random sample of size n from a normal distribution with mean the rv has a probability distribution called a t distribution with n – 1 degrees of freedom (df).

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Properties of t Distributions Let t v denote the density function curve for v df. 1. Each t v curve is bell-shaped and centered at Each t v curve is spread out more than the standard normal (z) curve.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Properties of t Distributions 3. As v increases, the spread of the corresponding t v curve decreases. 4. As, the sequence of t v curves approaches the standard normal curve (the z curve is called a t curve with df =

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. t Critical Value Let = the number on the measurement axis for which the area under the t curve with v df to the right of is called a t critical value.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Pictorial Definition of 0

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval Let and s be the sample mean and standard deviation computed from the results of a random sample from a normal population with mean The

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 7.3 to 7.5 Tests of Hypotheses

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Tests of Hypotheses Based on a Single Sample

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypotheses and Test Procedures

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypotheses The null hypothesis, denoted H 0, is the claim that is initially assumed to be true. The alternative hypothesis, denoted by H a, is the assertion that is contrary to H 0. Possible conclusions from hypothesis- testing analysis are reject H 0 or fail to reject H 0.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. A Test of hypotheses A test of hypotheses is a method for using sample data to decide whether the null hypothesis should be rejected.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Test procedure A test procedure is specified by 1.A test statistic, a function of the sample data on which the decision is to be based. 2.A rejection region, the set of all test statistic values for which H 0 will be rejected (null hypothesis rejected iff the test statistic value falls in this region.)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Errors in hypothesis testing A type I error consists of rejecting the null hypothesis H 0 when it was true. A type II error involves not rejecting H 0 when H 0 is false.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Rejection region: Suppose an experiment and a sample size are fixed, and a test statistic is chosen. The decreasing the size of the rejection region to obtain a smaller value of results in a larger value of for any particular parameter value consistent with H a.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Significance level Specify the largest value of that can be tolerated and find a rejection region having that value of. This makes as small as possible subject to the bound on. The resulting value of is referred to as the significance level.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Level test A test corresponding to the significance level is called a level test. A test with significance level is one for which the type I error probability is controlled at the specified level.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Tests About a Population Mean

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Case I: A normal population with known Null hypothesis: Test statistic value:

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Case I: A normal population with known Alternative Hypothesis Rejection Region for Level Test or

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Recommended steps in hypothesis-testing analysis 1.Identify the parameter of interest and describe it in the context of the problem situation. 2.Determine the null value and state the null hypothesis. 3.State the alternative hypothesis.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypothesis-testing analysis 4.Give the formula for the computed value of the test statistic. 5.State the rejection region for the selected significance level 6.Compute any necessary sample quantities, substitute into the formula for the test statistic value, and compute that value.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypothesis-testing analysis 7. Decide whether H 0 should be rejected and state this conclusion in the problem context. The formulation of hypotheses (steps 2 and 3) should be done before examining the data.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Case II: Large-sample tests When the sample size is large, the z tests for case I are modified to yield valid test procedures without requiring either a normal population distribution or a known

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large sample tests (n > 30) For large n, s is close to Test Statistic: The use of rejection regions for case I results in a test procedure for which the significance level is approximately

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Case III: A normal population distribution If X 1,…,X n is a random sample from a normal distribution, the standardized variable has a t distribution with n – 1 degrees of freedom.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The one-sample t- test Null hypothesis: Test statistic value:

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alternative Hypothesis Rejection Region for Level Test or The one-sample t -test

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P - Values

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P - value The P-value is the smallest level of significance at which H 0 would be rejected when a specified test procedure is used on a given data set.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P - value The P-value is the probability, calculated assuming H 0 is true, of obtaining a test statistic value at least as contradictory to H 0 as the value that actually resulted. The smaller the P- value, the more contradictory is the data to H 0.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P-values for a z -test P-value: upper-tailed test lower-tailed test two-tailed test

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P-value (area) z -z z Upper-Tailed Lower-Tailed Two-Tailed

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P–values for t -tests The P-value for a t test will be a t curve area. The number of df for the one-sample t test is n – 1.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 7.6 Relation between Confidence Interval and Testing Hypotheses