1. Statistical Fundamentals: Using Microsoft Excel for Univariate and Bivariate Analysis Alfred P. Rovai Estimation PowerPoint Prepared by Alfred P. Rovai Presentation © 2013 by Alfred P. Rovai Microsoft® Excel® Screen Prints Courtesy of Microsoft Corporation.

2. Estimation Estimation is a way to estimate a population parameter based on measuring a sample. It can be expressed in two ways A point estimate of a population parameter is a single value of a statistic, e.g., the sample mean is a point estimate of the population mean μ An interval estimate, e.g., confidence interval, is defined by two numbers, between which a population parameter lies Presentation © 2013 by Alfred P. Rovai

3. Point Estimates vs. Interval Estimates Polling is a common method of estimating population parameters Sample mean x̄ is the best point estimate of the population mean μ Sample proportion p of x successes in a random sample of n observations is the best point estimate of the population proportion p However, point estimates provide no measure of reliability Confidence intervals, on the other hand, provide a confidence level Presentation © 2013 by Alfred P. Rovai ^ http://abcnews.go.com/US/PollVault/

4. Estimating Confidence Intervals A confidence interval is an estimated range of values that is likely to include an unknown population parameter Confidence intervals are constructed at a confidence level, e.g., 95%, selected by the researcher – If a population is sampled repeatedly and interval estimates are made on each occasion, the resulting intervals will reflect the true population parameter in approximately 95% of the cases – This example corresponds to hypothesis testing with p =.05 Presentation © 2013 by Alfred P. Rovai

5. Steps for Calculating the Confidence Interval for an Unknown Population Parameter 1 Obtain the point estimate of the parameter. This is usually the sample mean or sample proportion. 2 Select a confidence level, e.g., 95% (alpha =.05) 3 Calculate the confidence interval for the unknown population parameter Presentation © 2013 by Alfred P. Rovai

6. Calculating the Confidence Interval (CI) for μ When σ Is Known Assumption Population σ and sample x̄ are known General formulas CI = Point Estimate ± Margin of Error (i.e., Sampling Error) CI = x̄ ± (Critical Value)*(Standard Error) Calculating formula or where C = critical value for the required CI in standard deviation units (z-scores) Presentation © 2013 by Alfred P. Rovai

7. Critical Values Use the normal distribution to calculate critical values 90% CI =NORM.S.INV(1-0.10/2) = 1.645 (90% of the area of a normal distribution is within 1.96 standard deviations of the mean) 95% CI =NORM.S.INV(1-0.05/2) = 1.96 (95% of the area of a normal distribution is within 1.96 standard deviations of the mean) 99% CI =NORM.S.INV(1-0.01/2) = 2.58 (99% of the area of a normal distribution is within 2.58 standard deviations of the mean) Presentation © 2013 by Alfred P. Rovai

8. Example: 95% CI, n = 100, σ = 10 95% CI = 50 ± 1.96*(10/√100) = 50 ± 1.96 = 48.04, 51.96 Margin of error =CONFIDENCE.NORM(alpha,standard_dev,size) =CONFIDENCE.NORM(0.05,10,100) = 1.96 Presentation © 2013 by Alfred P. Rovai

9. The margin of error for the previous example is 1.96 units. What is the required sample size to be 95% confident that the estimate is within 1 unit of the true mean? Solution The required sample is 385 Presentation © 2013 by Alfred P. Rovai Example Continued

10. Example: 95% CI, n = 100, σ = 10 We are 95% confident that the true population mean is between 48.04 and 51.96 Although we cannot be certain (i.e., 100% confident) that the true mean is in this interval, 95% of intervals formed by taking random samples from the target population in this manner will contain the true mean Presentation © 2013 by Alfred P. Rovai

11. If the population standard deviation σ is unknown, use the sample standard deviation S in calculating CI – This procedures increases uncertainty, since S varies from sample to sample Use the student ’ s t distribution instead of the normal Z distribution to calculate margin of error Margin of Error =CONFIDENCE.T(alpha,standard_dev,size) where size = sample size Presentation © 2013 by Alfred P. Rovai Calculating the Confidence Interval (CI) for μ When σ Is Unknown

12. Sample proportion p = x/n is the best point estimate of the population proportion p where x = number of successes in sample size n 95% CI for p Presentation © 2013 by Alfred P. Rovai Calculating the Confidence Interval (CI) for an Unknown Population Proportion p ^

13. Question: Overall, how much do you feel you can trust the government in Washington to do what’s right? Reported Poll Results 95% CI Calculation Can trust = 39, n = 39 + 60 + 1 = 100, p = 39/100 =.39 Therefore, the interval (.295,.485) captures p 95% of the time Presentation © 2013 by Alfred P. Rovai Example ^

14. The margin of error for the previous example is 9.53%. What is the required sample size to be 95% confident that the estimate is within 3% of the correct percentage? Solution The required sample is 1068 Presentation © 2013 by Alfred P. Rovai Example Continued

15. Commonly used confidence level multipliers – 99% confidence level multiplier = 2.58 – 95% confidence level multiplier = 1.96 – 90% confidence level multiplier = 1.645 The higher the confidence level, the wider the CI Increasing the random sample of n observations will make a CI with the same confidence level narrower (i.e., more precise) Presentation © 2013 by Alfred P. Rovai Summary

16. Estimation End of Presentation Copyright 2013 by Alfred P. Rovai