1 BA 275 Quantitative Business Methods Confidence Interval Estimation Estimating the Population Proportion Hypothesis Testing Elements of a Test Concept behind a Test Examples Agenda

2 Quiz #4 : Part I An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with mean of 117 cm and standard deviation of 5.2 cm. Find the probability that one selected subcomponent is longer than 120 cm. Find the probability that if four subcomponents are randomly selected, their mean length exceeds 120 cm. You, as an inspector, randomly selected 25 subcomponents and measured their mean length to be 120 cm. Based on your sample, do you think the machine is still in control (i.e., the machine is still producing subcomponents with a mean length of 117 cm), or is already out of control (the mean length is no longer 117 cm as required)? Support your argument with numerical evidence.

3 Quiz #4: Part II The mean amount for all of the invoices for your company last month is not known. Based on your past experience, you are willing to assume that the standard deviation of invoice amounts is about $200. If you take a random sample of 100 invoices, what is the value of the standard deviation for the sample mean ? The empirical rule says that the probability is about 0.95 that the sample mean is within $A of the population mean m. What is the value of A? Construct a 95% confidence interval for the true population mean.

4 Central Limit Theorem In the case of sample mean In the case of sample proportion

5 100(1-  )% Confidence Interval for the Proportion (p.13, formula 3)

6 Example (p.14) A sample of 35 student information sheets shows that 9 intend to concentrate in Finance. Give a 99% confidence interval for the proportion of students in the population that intend to concentrate in Finance.

7 Margin of Error (B) Margin of Error (how good is your point estimate?)

8 Estimation in Practice Determine a confidence level (say, 95%). How good do you want the estimate to be? (define margin of error) Use formulas (p.14) to find out a sample size that satisfies pre-determined confidence level and margin of error.

9 Example (p.14) A marketing manager for a start-up firm in Michigan wishes to discover the proportion of teenagers in Japan who own a CD player. If the manager wants a confidence interval of width 0.1, how many teenagers must be sampled? Use a conservative estimate of p. Assume that the confidence level is to be 95%.

10 AccuracyAccuracy Gained by Increasing the Sample Size Margin of Error (B)Sample Size (n) 7%196 6%266 5%384 4%600 3%1037 2%2401 1%9604

11 Statistical Inference: Estimation Research Question: What is the parameter value? Sample of size n Population Tools (i.e., formulas): Point Estimator Interval Estimator Example:  = 10,000 n = 100 What is the value of  ? Example:  ?

12 Statistical Inference: Hypothesis Testing Research Question: Is a claim about the parameter value supported? Sample of size n Population Tool (i.e., formula): Z or T score Example:  = 10,000 n = 100 Is “  > 22,000”? Example: “  22,000”?

13 Elements of a Test Hypotheses Null Hypothesis H 0 Alternative Hypothesis H a Test Statistic Decision Rule (Rejection Region) Before collecting data After collecting data Evidence (actual observed test statistic) Conclusion Reject H 0 if the evidence falls in the R.R. Do not reject H 0 if the evidence falls outside the R.R.

14 Example A bank has set up a customer service goal that the mean waiting time for its customers will be less than 2 minutes. The bank randomly samples 30 customers and finds that the sample mean is 100 seconds. Assuming that the sample is from a normal distribution and the standard deviation is 28 seconds, can the bank safely conclude that the population mean waiting time is less than 2 minutes?