1. Confidence Intervals for Proportions Chapter 8, Section 3 Statistical Methods II QM 3620

2. Something Changed Quantitative variables vs. Qualitative variables Sometimes we need to make an estimation about a qualitative variable Whether or not a customer is satisfied with your company’s services is a qualitative variable. They could answer “Yes” or “No”, or the answer could be “Very Satisfied” “Somewhat Satisfied”, etc. For accountants, whether or not an account balance is correct is qualitative. The actual amount it is off would be quantitative, but the status would be qualitative.

3. Something Changed In fact, it is likely that more of the variables you encounter are qualitative than quantitative (gender, ethnicity, income level, etc.) But income is quantitative, right? That is a number, not a category Yes, but we can make a quantitative variable qualitative by reducing the scale level of the data We could convert a quantitative variable to a qualitative variable by putting the variable in categories. Income between $30K and $40K, etc.

4. The Population Box Suppose that this box is full of our customers. Let’s say that the ones shaded in red are satisfied with our service. We could simply attempt to talk to ALL of these customers, but that could be difficult and expensive. Or we could take a sample and use it to infer about all the members of the population “box”, whether we actually talk to them or not.

5. The Sample So, instead of talking to all of the customers, we take a sample from the “population box” to estimate the proportion of red individuals (those who are satisfied with our service). But … remember the problems with samples. Each time we take a sample, we are likely to end up with a different proportions of “red” individuals. We will need to take this sample variation into account.

6. p =.49 p =.55 p =.52 p =.48 p =.57 Sampling Distribution We know that if we take a sample, the proportion of“red” individuals will vary from sample to sample. How much these sample proportions vary is measured by their standard error. We refer to the population proportion as  and the proportion in a sample as p.  =?

7. Is it Magic? So, our purpose for this module is to take ONLY ONE SAMPLE from a closed box and us it to estimate the proportion of, in this case, “red” individuals in the box. How do we do that? Another confidence interval!!

8. Another Confidence Interval Confidence intervals can be used to estimate pretty much any population parameter. Just keep in mind that a confidence interval is used to estimate a number in a larger group by only looking at the limited information provided by a sample of that group. The only extension we are doing in this module is the application of a confidence interval to a population proportion. This allows us to analyze qualitative values (like gender) versus quantitative variables (like age). So what is different?

9. Another Confidence Interval Well, the calculations are different because you now have to count the observations rather than average them. The multiple in the confidence interval formula is now based on the z-score rather than the t-value. Read the discussion of interval estimation for proportions and their calculation (pages 358-359). The best part about confidence intervals for proportions is that we need such little information. All we need is the sample size (easy, just count), the proportion that fell into the category we are interested in (almost as easy, just count and divide by the sample size), and …

10. The Equation The general formula for all confidence intervals is: Point Estimate  Multiple Standard Error of Point Estimate  The version for a confidence interval for a population proportion is: Sample Proportion  z-value Standard Error of Sample Proportion  The calculation to the right hand side of the plus-minus sign (±) is still the margin of error, but in this case it is the margin of error for an estimate of a population proportion.

11. The Equation The standard error of the sample proportion can again be estimated directly from the sample you took. Sample proportions near 50% create the widest confidence intervals because half belong to one group and half belong to the other group. If the sample is 99% one group and 1% the other group, there will be a lot less variation among the sample members. Like all confidence intervals, more information (i.e. larger sample sizes) lead to smaller intervals. The multiple only handles the confidence level. Greater confidence requires a larger multiple … and subsequently a wider interval.

12. Now based on a z-score. The good news is that you have seen z-scores before and should have some familiarity. The bad news is that the function to look up z-scores in Excel requires some practice. I do expect you to be able to find z-scores using Excel, not a table in the book (and don’t think I can’t ask you for one that is not in the table.) The Multiple

13. Finding a z value using Excel The z Distribution To find a z value in Excel, you use the NORMSINV function. The NORMSINV function needs only one bit of information: 1) The confidence level you want + z value So, a z value for a 95% confidence (or.95 in decimal format) would be: =NORMSINV(.95+(1-.95)/2); or = NORMSINV(.975) This is where it gets messy in Excel. To get the proper z value for a confidence level, you need to include the lower part of the“unconfidence” region in the function. For example,to look up the z value for a 95% confidence interval, you would take the 95% and add half of the “unconfidence“ of 5% or 2.5%. That means the NORMSINV function looks up 97.5% (or.975) for a 95% confidence interval. You only want the middle 95%,but the Excel function won’t provide it that way. For a 99% confidence interval,you would used NORMSINV(.995). For 90% confidence, you would use NORMSINV(.95). The NORMSINV function looks up the blue area in the graph above. 95%2.5% 97.5% total

14. Putting it all together … We have a point estimate … the proportion we calculated from the sample … referred to as p We have the sample size … the number of observations in the sample … which is referred to as n We have the multiple … the z-value from Excel … which is referred to as z That gives us everything we need to calculate a confidence interval for a proportion. We have everything we need

15. z-value  Sample Proportion The Calculation Remember our confidence interval equation from an earlier slide, Sample Proportion  Standard Error of Algebraically, the equation looks like: Where and z  z-value p(1  p) n  Standard Error of Sample Proportion p  Sample Proportion p  1 - p  n p zp z

16. Factors Affecting Margin of Error  Sample proportion ( p ): Sample size: Confidence Level ( CL ): as p .50 as n as CL Margin of Error p  1-p  n  z z … and margin of error is directly related to precision. A smaller margin of error is a more precise estimate. Margin of Error Margin of Error Margin of Error

17. Remember, our whole reason for this series of thoughts and calculations was to help us best estimate the proportion with some characteristic in a large group by using only the limited information provided by a sample from that group. SoWhat was the Point of All This?

18. The Point Estimate The point estimate for the population proportion is the sample proportion. Generally, when you want to estimate something in a population, use the same calculation in the sample … generally. The Confidence Level Confidence levels mean the same thing regardless of what you are estimating. A 95% confidence level means that 19 out of 20 times you will have an accurate interval. Using the point estimate alone will pretty much guarantee that you are incorrect. Precision or accuracy – your choice. Using Statistics

19. The Standard Error of the Point Estimate The sample proportion, which is our point estimate, is going to change from sample to sample. We are going to take this into account when forming our confidence interval. The Assumptions The sample size is going to need to be large enough to justify our use of the normal distribution to estimate the proportion. This will be true when the number of observations that fall into each group is at least 5. So if you are estimating the proportion of a certain characteristic in a population, make sure that the number of observations in your sample that have the characteristic is at least 5, and the number of observations that does not have the characteristic is at least 5. Using Statistics

20. Let’s try this for real ApplicationTime

21. Business application on page 360 - Quick Lube operates oil-change outlets. A computer is used to send out reminder cards to customers 3 months after a service. Management is interested in the success rate of this reminder system in bringing back customers. A sample of 100 customers is evaluated and of the 100, 62 returned within one month of receiving the reminder card. BusinessApplication Highlights

22. ANOTHER EXAMPLE

23. A multinational corporation would like to estimate the proportion of its workers who currently commute to work by using a carpool. The corporation hopes to develop a proposal to encourage more employees to forgo their automobile as means of maker the company “greener” A random sample of 156 employees was taken from company records and the most-often used transportation method to work of these employees was recorded. Business Application Highlights

24. The responses were separated into one of three categories: carpool the employee’s own vehicle public transportation Management has asked for an estimate at the 90% confidence level Business Application Highlights