1. Welcome to MM207 - Statistics! Unit 6 Seminar: Inferential Statistics and Confidence Intervals

2. Definition Review Population - a set of measurements Parameters described the characteristics of a population. Sample: a subset of measurements from the population Statistics describe the characteristics of a sample. Most of the time we do not have the entire population, we have a sample from the population. Therefore, we must use sample statistics to estimate population parameters. We use a confidence interval to estimate a population mean or a proportion.

3. Confidence Intervals for μ or p There are two steps 1.Find E (MoE or margin of error). 2. Find the interval.

4. Critical Formula σ Known Margin of Error [E] E = z c (σ/√n); the level of confidence is determined by the value selected for z c C-Confidence Interval xbar – E < µ < xbar + E Minimum Sample Size N = (z c* σ/E)^2

5. Step 2: Compute the Interval The interval has a lower number and an upper number For estimating μ x bar – E < μ < x bar + E For estimating p p hat – E < p < p hat + E

6. Example 1: CI for μ, n ≥ 30 n = 40 xbar = 12 σ = 5 Find the 95% CI for μ. Step 1: Find E Step 2: Find the interval Since n ≥ 30, σ known x bar – E < μ < x bar + E E = z c * σ / √[n] 12 – 1.55 < μ < 12 + 1.55 E = 1.96 * 5 / √[40] 10.45 < μ < 13.55 E = 9.8 / 6.32455532 E ≈ 1.549516054 ≈ 1.55 Use the t-table, the bottom row, to find z c = 1.96 Or use CONFIDENCE in Excel to find E

7. Excel for Confidence Interval Sigma Known (z) Alpha is the complement of the confidence interval, this is for the 80% confidence interval This is E

8. Critical Formula Small Samples t-Distribution –t = [xbar - µ] / s/√n Margin of Error [E] – E = t c (s/√n); the level of confidence is determined by the value selected for z c C-Confidence Interval –xbar – E < µ < xbar + E Minimum Sample Size –N = (t c* s/E)^2

9. Example 2: CI for μ, n < 30 n = 20 df = 19 xbar = 12 s = 5 Find the 95% CI for μ. Step 1: Find E Step 2: Find the interval n < 30, σ not known x bar – E < μ < x bar + E df = 19 12 – 2.34 < μ < 12 + 2.34 E = t c * s / √[n] 9.66 < μ < 14.34 E = 2.093 * 5 / √[20] E = 10.465 / 4.472135955 E ≈ 2.340045138 ≈ 2.34 Use the t-table, df = 19, to find 2.093

10. Excel for Confidence Interval Small Samples (t) Score Mean13 Standard Error1.724819 Median11.5 Mode6 Standard Deviation6.899275 Sample Variance47.6 Kurtosis-0.21368 Skewness0.77674 Range23 Minimum5 Maximum28 Sum208 Count16 Confidence Level(99.0%)5.082546 This is the function that will give you E using the t distribution E

11. z-Estimate of a Proportion Sample proportion 0.3333 Sample size 300 Confidence level 0.99 Confidence Interval Estimate 0.3333 +/- 0.0701 Lower confidence limit 0.2632 Upper confidence limit 0.4034 This is a home grown procedure. Enter the data on the left. The answers will be shown in Red.

12. Example 3: CI for p n = 400 p hat = 0.6, q hat = 1 – 0.6 = 0.4 Find the 95% CI for p. np hat = 240 > 5, nq hat = 160 > 5, ok to use z c Step 1: Find EStep 2: Find the interval E = z c * √[pq / n] p hat – E < p < p hat + E E = 1.96 * √ [(0.6 * 0.4) / 400] 0.6 – 0.048 < p < 0.6 + 0.048 E = 1.96 * √ [0.24 / 400] 0.552 < p < 0.648 E = 1.96 *.024494897 E ≈ 0.048009998 ≈ 0.048

13. Example 4: Choosing the Normal or t-Distribution Page 329, using the flow chart n = 25 σ = $28,000 x bar = $181,000 Normal or t-Distribution (z c or t c )? n = 18 s = $24,000 x bar = $162,000 Normal or t-Distribution?

14. Other Topics Finding a minimum sample size for a confidence interval Finding z c for a confidence level Interpreting a confidence interval Comparing confidence intervals for a level of 90%, 95%, and 99%

15. Finding a minimum sample size for a confidence interval Page 316 Find n for a 99% CI given σ ≈ s ≈ 10 and E = 3.2 n = [(z c * σ) / E] 2 n = [2.575* 10 / 3.2] 2 n = [25.75 / 3.2] 2 n = [8.046875] 2 n = 64.75 or 65 Note: Always round up! For example, you would round 72.1 to 73 because we need at least 72.1 for the sample size.

16. Finding Zc for a Confidence Level Sometimes the z c for the confidence level is not provided in a table. Find the z c for an 85% CI. This z c is not in the t-table. 1/2(1 - 0.85) = 0.15/2 = 0.075 Find the z for 0.0750 in the Standard Normal Table z c = - 1.44 or z c = 1.44 Note: Use the positive z c in the formula for E.

17. Interpreting a Confidence Interval Example 1. The interval we found is 10.45 < μ < 13.55 With 95% confidence, we can say that the population mean is between 10.45 and 13.55. Example 2. The interval we found is 9.66 < μ < 14.34 With 95% confidence, we can say that the population mean is between 9.66 and 14.34. Example 3. The interval we found is 0.552 < p < 0.648 With 95% confidence, we can say that the population proportion is between 55.2% and 64.8%.

18. Comparing confidence intervals for a level of 90%, 95%, and 99% n = 40 xbar = 12 σ = 5 For the 90% CI, E ≈ 1.30 and the interval is 10.70 < μ < 13.30 For the 95% CI, E ≈ 1.55 and the interval is 10.45 < μ < 13.55 For the 99% CI, E ≈ 2.04 and the interval is 9.96 < μ < 14.04 As the confidence level increases, the interval width increases. We have greater confidence, but less precision in estimating μ.

19. Have a great week!