Week 7 October 13-17

1. Week 7 October 13-17 Three Mini-Lectures QMM 510 Fall 2014

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

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

4. 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: Rule: It is safe to assume normality of p = x/n if we have at least 10 “successes” and 10 “failures” in the sample. Table 8.9

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

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

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

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

10. Sample Size Determination for a Mean Chapter 8 Using MegaStat 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:

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

12. 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 SampleTake 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 DataHow often are such samples available? p might be different enough to make it a questionable assumption. Sample Size Determination for a Mean Chapter 8 How to Estimate p?

13. Sample Size Determination for a Mean Chapter 8 Using MegaStat 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)?.

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 H0and H1. LO9-3:Define Type I error, Type II error, and power. LO9-4: Formulate a null and alternative hypothesis for μ or π.

15. Logic of Hypothesis Testing Chapter 9

16. Logic of Hypothesis Testing Chapter 9 LO9-2: Explain the difference between H0 and H1. • 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. • H0: Null hypothesisH1: Alternative hypothesis • These two statements are hypotheses because the truth is unknown. State the Hypothesis

17. Logic of Hypothesis Testing Chapter 9 State the Hypothesis • Efforts will be made to reject the null hypothesis. • If H0 is rejected, we tentatively conclude H1 to be the case. • H0 is sometimes called the maintained hypothesis. • H1 is called the action alternative because action may be required if we reject H0 in favor of H1. Can Hypotheses Be Proved? • We cannot accept a null hypothesis; we can only fail to reject it. Role of Evidence • The null hypothesis is assumed true and a contradiction is sought.

18. Logic of Hypothesis Testing Chapter 9 LO9-3: Define Type I error, Type II error, and power. Types of Error • Type I error: Rejecting the null hypothesis when it is true. This occurs with probability a (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 b. Also called a false negative.

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

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

21. Logic of Hypothesis Testing Chapter 9 Relationship between a and b • Both a small a and a small b are desirable. • For a given type of test and fixed sample size, there is a trade-off between a and b. • The larger critical value needed to reduce a risk makes it harder to reject H0, thereby increasing b risk. • Both a and b can be reduced simultaneously only by increasing the sample size.

22. Logic of Hypothesis Testing Chapter 9 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.)

23. Statistical Hypothesis Testing Chapter 9 LO9-4: Formulate a null and alternative hypothesis for μ or π. • A statistical hypothesisis a statement about the value of a population parameter. • A hypothesis testis 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.

24. Statistical Hypothesis Testing Chapter 9 One-Tailed and Two-Tailed Tests • The direction of the test is indicated by H1: > indicates a right-tailed test < indicates a left-tailed test ≠ indicates a two-tailed test

25. Statistical Hypothesis Testing Chapter 9 Decision Rule • A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error. • The decision ruleuses the known sampling distribution of the test statistic to establish the critical valuethat divides the sampling distribution into two regions. • Reject H0 if the test statistic lies in the rejection region.

26. Statistical Hypothesis Testing Chapter 9 Decision Rule for Two-Tailed Test • Reject H0 if the test statistic < left-tail critical value or if the test statistic > right-tail critical value.

27. Statistical Hypothesis Testing Chapter 9 When to use a One- or Two-Sided Test • A two-sided hypothesis test (i.e., µ ≠ µ0) is used when direction (< or >) is of no interest to the decision maker. • A one-sided hypothesis test is used when - the consequences of rejecting H0 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.

28. Statistical Hypothesis Testing Chapter 9 Decision Rule for Left-Tailed Test • Reject H0 if the test statistic < left-tail critical value. Figure 9.2

29. Statistical Hypothesis Testing Chapter 9 Decision Rule for Right-Tailed Test • Reject H0 if the test statistic > right-tail critical value.

30. Statistical Hypothesis Testing Chapter 9 Type I Error also called a false positive • A reasonably small level of significance a is desirable, other things being equal. • Chosen in advance, common choices for a are .10, .05, .025, .01, and .005 (i.e., 10%, 5%, 2.5%, 1%, and .5%). • The a risk is the area under the tail(s) of the sampling distribution. • In a two-sided test, the a risk is split with a/2 in each tail since there are two ways to reject H0.