Summary of Class Discussion and Homework For Statistics for Business Control, etc. STAT-UB.0103.003 3 April 2017 Ed Melnick Class Discussion: 1. Statistical Inference A. Estimation 1. Point estimation 2. Interval estimation Document 20 (page 127) represents the transition from probability to statistics. The exercise consists of a set of examples. a. Probability 1 Probability over an interval when the mean, variance, and distribution are known. 2 Probability over an interval when the mean and variance are known and the Central Limit Theorem is used to get an asymptotic distribution since the underlying probability function is unknown. b. Statistics 3 Confidence interval for the mean when the variance is known, and the underlying distribution is unknown. 4 Confidence interval for the mean when the variance and the underlying distribution are unknown. 2. Hypothesis Testing A. Important terms: 1. Assumptions 2. Hypotheses - null and alternative 3. Hypotheses - simple and composite 4. Type I (significance level) and Type II errors 5. Critical regions 6. One and two tail tests - as an aside, in courts of law, all tests are two tail tests even if the question only deals with a one sided issue. This is to minimize gaming in trials. B. Hypothesis testing leads to either a rejection of fail to reject (practical terms, to accept) the null hypothesis. If the null hypothesis is rejected, the alternative hypothesis is treated as correct. C. The relationship between hypothesis testing and confidence interval was demonstrated. D. Problems with hypothesis testing 1. Binary decision making - reject or fail to reject the null hypothesis 2. All hypotheses are rejected for sufficiently large samples 3. Difficulty stating the alternative hypothesis E. Most useful when developing a model; i.e., determining the usefulness of incremental changes. Take Aways 1. Use statistical methodologies that describe the problem being solved - check assumptions. 2. Check solutions for internal consistency. 3. Sample based estimators are random variables that are described by a probability function and are represented with Latin letters. Population parameters are constants and are represented with Greek letters. 4. Estimates of population parameters should be embedded in a confidence interval. 5. The sample size determination formula provides the minimum sample size required to bound the sample estimation error based on a given level of probability Homework Due 10 April 1. Read Chapter 6 (skipping section 6.7) in MBS on confidence intervals. 2. Read Chapter 7 (skipping section 7.7) in MBS on hypothesis testing. 3. Read Section 2.8 (bivariate distribution) and Section 11.5 (correlation) in MBS. 4. Below is a modified rewording of Assignment 19 in the notes. Solve these problems. Note that in question 1 the population variance is not known and each datum in question 2 is either error or non-error in an accounting statement (cost accounting problem). APPLICATIONS OF SIGNIFICANCE TESTING 1. Estimation of Mean Monthly Rentals a. Assume a sample of 10 monthly rentals from a large housing complex of 300,000 units is: 980, 1230, 1320, 1440, 1180, 1050, 1520, 1630, 880, 920. Suppose the landlord association claims that the true average rental of the low income 3 room apartment is $1100. The housing authority believes that the average rental is higher. State and test the null hypothesis of the landlord association's claim versus the alternative hypothesis of the housing authority claim at the = 0.05 significance level. Does the data support the landlord's association claim? What would you report to the housing authority? b. Suppose that a sample of 10,000 apartments from a much larger complex of apartments, with identical characteristics as the complex described in question 1, yields the following results: X 1110, s 250. Using this data, what are your conclusions? Use = 0.05 (or your favorite type I error). What would you report to the housing authority? 2. Applications to Auditing a. Suppose it is claimed that more efficient record keeping has resulted in a lower error rate in accounting entries than the 3% which has been true in the past. The null hypothesis is that the error rate is .03; the alternate hypothesis is that the error rate is greater than 0.03. In a sample of 15 entries, 2 errors were found. Does the data support the claim in the null hypothesis at the = 0.05 significance level? b. In a large study of 1000 entries, 40 errors were found. Does the data support the null hypothesis at the = 0.05 significance level