Week 7 October Three Mini-Lectures QMM 510 Fall 2014

8-2 A proportion is a mean of data whose only values are 0 or 1. Chapter 8 Confidence Interval ML 7.1 For a Proportion (  )

8-3 The distribution of a sample proportion p = x/n is symmetric if  =.50. The distribution of p approaches normal as n increases, for any  Applying the CLT Applying the CLT Chapter 8 Confidence Interval for a Proportion (  )

8-4 Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both n   10 and n(1  )  10. When Is It Safe to Assume Normality of p? When Is It Safe to Assume Normality of p? Sample size to assume normality: Table 8.9 Chapter 8 Confidence Interval for a Proportion (  ) Rule: It is safe to assume normality of p = x/n if we have at least 10 “successes” and 10 “failures” in the sample.

8-5 Confidence Interval for  Confidence Interval for  The confidence interval for the unknown  (assuming a large sample) is based on the sample proportion p = x/n.The confidence interval for the unknown  (assuming a large sample) is based on the sample proportion p = x/n. Chapter 8 Confidence Interval for a Proportion (  )

8-6 Example: Auditing Example: Auditing Chapter 8 Confidence Interval for a Proportion (  )

8-7 Chapter 8 Estimating from Finite Population N = population size; n = sample size The FPCF narrows the confidence interval somewhat. When the sample is small relative to the population, the FPCF has little effect. If n/N <.05, it is reasonable to omit it (FPCF  1 ).

8-8 To estimate a population mean with a precision of + E (allowable error), you would need a sample of size n. Now, Sample Size to Estimate  Sample Size to Estimate  Chapter 8 Sample Size Determination ML 7-2

8-9 Method 1: Take a Preliminary Sample Take a small preliminary sample and use the sample s in place of  in the sample size formula. Method 2: Assume Uniform Population Estimate rough upper and lower limits a and b and set  = [(b  a)/12] ½. How to Estimate  ? How to Estimate  ? Chapter 8 Method 3: Assume Normal Population Estimate rough upper and lower limits a and b and set  = (b  a)/4. This assumes normality with most of the data within  ± 2  so the range is 4 . Method 4: Poisson Arrivals In the special case when  is a Poisson arrival rate, then  =   Sample Size Determination for a Mean

8-10 Using MegaStat Using MegaStat Chapter 8 Sample Size Determination for a Mean For example, how large a sample is needed to estimate the population mean age of college professors with 95 percent confidence and precision of ± 2 years, assuming a range of 25 to 70 years (i.e., 2 years allowable error)? To estimate σ, we assume a uniform distribution of ages from 25 to 70:

8-11 To estimate a population proportion with a precision of ± E (allowable error), you would need a sample of size n. Since  is a number between 0 and 1, the allowable error E is also between 0 and 1. Chapter 8 Sample Size Determination for a Mean

8-12 Method 1: Assume that  =.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  in the sample size formula. Method 3: Use a Prior Sample or Historical Data How often are such samples available?  might be different enough to make it a questionable assumption. How to Estimate  ? How to Estimate  ? Chapter 8 Sample Size Determination for a Mean

8-13 Using MegaStat Using MegaStat Chapter 8 Sample Size Determination for a Mean For example, how large a sample is needed to estimate the population proportion with 95 percent confidence and precision of ±.02 (i.e., 2% allowable error)?.

9-14 One-Sample Hypothesis Tests ML 7-3 Chapter 9 Learning Objectives LO9-1: List the steps in testing hypotheses. LO9-2: Explain the difference between H 0 and H 1. LO9-3: Define Type I error, Type II error, and power. LO9-4: Formulate a null and alternative hypothesis for μ or π.

9-15 Chapter 9 Logic of Hypothesis Testing

9-16 Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about some fact about a population. One statement or the other must be true, but they cannot both be true. H 0 : Null hypothesis H 1 : Alternative hypothesis These two statements are hypotheses because the truth is unknown. State the Hypothesis State the Hypothesis Chapter 9 LO9-2: Explain the difference between H 0 and H 1. Logic of Hypothesis Testing

9-17 Efforts will be made to reject the null hypothesis. If H 0 is rejected, we tentatively conclude H 1 to be the case. H 0 is sometimes called the maintained hypothesis. H 1.H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1. State the Hypothesis State the Hypothesis Chapter 9 Can Hypotheses Be Proved? Can Hypotheses Be Proved? We cannot accept a null hypothesis; we can only fail to reject it.We cannot accept a null hypothesis; we can only fail to reject it. Role of Evidence Role of Evidence The null hypothesis is assumed true and a contradiction is sought. Logic of Hypothesis Testing

9-18 Types of Error Types of Error Type I error: Rejecting the null hypothesis when it is true. This occurs with probability  (level of significance). Also called a false positive.Type I error: Rejecting the null hypothesis when it is true. This occurs with probability  (level of significance). Also called a false positive. Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability . Also called a false negative.Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability . Also called a false negative. Chapter 9 LO9-3: Define Type I error, Type II error, and power. Logic of Hypothesis Testing

9-19 Probability of Type I and Type II Errors Probability of Type I and Type II Errors If we choose  =.05, we expect to commit a Type I error about 5 times in 100.If we choose  =.05, we expect to commit a Type I error about 5 times in 100.  cannot be chosen in advance because it depends on  and the sample size.  cannot be chosen in advance because it depends on  and the sample size. A small  is desirable, other things being equal.A small  is desirable, other things being equal. Chapter 9 Logic of Hypothesis Testing

9-20 Power of a Test Power of a Test Chapter 9 A low  risk means high power.A low  risk means high power. Larger samples lead to increased power.Larger samples lead to increased power. Logic of Hypothesis Testing

9-21 Relationship between  and  Relationship between  and  Both a small  and a small  are desirable.Both a small  and a small  are desirable. For a given type of test and fixed sample size, there is a trade-off between  and .For a given type of test and fixed sample size, there is a trade-off between  and . The larger critical value needed to reduce  risk makes it harder to reject H 0, thereby increasing  risk.The larger critical value needed to reduce  risk makes it harder to reject H 0, thereby increasing  risk. Both  and  can be reduced simultaneously only by increasing the sample size.Both  and  can be reduced simultaneously only by increasing the sample size. Chapter 9 Logic of Hypothesis Testing

9-22 Consequences of Type I and Type II Errors Consequences of Type I and Type II Errors The consequences of these two errors are quite different, and the costs are borne by different parties. Example: Type I error is convicting an innocent defendant, so the costs are borne by the defendant. Type II error is failing to convict a guilty defendant, so the costs are borne by society if the guilty person returns to the streets. Firms are increasingly wary of Type II error (failing to recall a product as soon as sample evidence begins to indicate potential problems.) Chapter 9 Logic of Hypothesis Testing

9-23 A statistical hypothesis is a statement about the value of a population parameter.A statistical hypothesis is a statement about the value of a population parameter. A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of the parameter.A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of the parameter. When testing a mean we can choose between three tests.When testing a mean we can choose between three tests. Chapter 9 Statistical Hypothesis Testing LO9-4: Formulate a null and alternative hypothesis for μ or π.

9-24 The direction of the test is indicated by H 1 :The direction of the test is indicated by H 1 : Chapter 9 > indicates a right-tailed test < indicates a left-tailed test ≠ indicates a two-tailed test Statistical Hypothesis Testing One-Tailed and Two-Tailed Tests One-Tailed and Two-Tailed Tests

9-25 Decision Rule Decision Rule A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error.A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error. The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions.The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions. Reject H 0 if the test statistic lies in the rejection region.Reject H 0 if the test statistic lies in the rejection region. Chapter 9 Statistical Hypothesis Testing

9-26 Decision Rule for Two-Tailed Test Decision Rule for Two-Tailed Test Reject H 0 if the test statistic right-tail critical value.Reject H 0 if the test statistic right-tail critical value. Chapter 9 Statistical Hypothesis Testing

9-27 When to use a One- or Two-Sided Test When to use a One- or Two-Sided Test A two-sided hypothesis test (i.e., µ ≠ µ 0 ) is used when direction ( ) is of no interest to the decision maker.A two-sided hypothesis test (i.e., µ ≠ µ 0 ) is used when direction ( ) is of no interest to the decision maker. A one-sided hypothesis test is used when - the consequences of rejecting H 0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher.A one-sided hypothesis test is used when - the consequences of rejecting H 0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher. Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal.Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal. Chapter 9 Statistical Hypothesis Testing

9-28 Decision Rule for Left-Tailed Test Decision Rule for Left-Tailed Test Reject H 0 if the test statistic < left-tail critical value.Reject H 0 if the test statistic < left-tail critical value. Figure 9.2 Chapter 9 Statistical Hypothesis Testing

9-29 Decision Rule for Right-Tailed Test Decision Rule for Right-Tailed Test Reject H 0 if the test statistic > right-tail critical value.Reject H 0 if the test statistic > right-tail critical value. Chapter 9 Statistical Hypothesis Testing

9-30 Type I Error Type I Error A reasonably small level of significance  is desirable, other things being equal.A reasonably small level of significance  is desirable, other things being equal. Chosen in advance, common choices for  areChosen in advance, common choices for  are.10,.05,.025,.01, and.005 (i.e., 10%, 5%, 2.5%, 1%, and.5%). The  risk is the area under the tail(s) of the sampling distribution.The  risk is the area under the tail(s) of the sampling distribution. In a two-sided test, the  risk is split with  /2 in each tail since there are two ways to reject H 0.In a two-sided test, the  risk is split with  /2 in each tail since there are two ways to reject H 0. Chapter 9 Statistical Hypothesis Testing false positive also called a false positive