1 / 19

Welcome to MM207 - Statistics!

Welcome to MM207 - Statistics!. Unit 6 Seminar: Inferential Statistics and Confidence Intervals. Definition Review. Population - a set of measurements Parameters described the characteristics of a population.

maris-holcomb
Download Presentation

Welcome to MM207 - Statistics!

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 populationStatistics 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 • Find E (MoE or margin of error). 2. Find the interval.

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

  5. Step 2: Compute the Interval The interval has a lower number and an upper number For estimating μ xbar – E < μ < xbar + E For estimating p phat – E < p < phat + 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 xbar – E < μ < xbar + E E = zc * 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 zc = 1.96 Or use CONFIDENCE in Excel to find E

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

  8. Critical Formula Small Samples • t-Distribution • t = [xbar - µ] / s/√n • Margin of Error [E] • E = tc (s/√n); the level of confidence is determined by the value selected for zc • C-Confidence Interval • xbar – E < µ < xbar + E • Minimum Sample Size • N = (tc*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 xbar – E < μ < xbar + E df = 19 12 – 2.34 < μ < 12 + 2.34 E =tc * 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 IntervalSmall Samples (t) E This is the function that will give you E using the t distribution

  11. z-Estimate of a Proportion • Sample proportion 0.3333 • Sample size 300 • Confidence level 0.99 • Confidence Interval Estimate0.3333 +/- 0.0701 • Lower confidence limit0.2632 • Upper confidence limit0.4034 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 phat = 0.6, qhat = 1 – 0.6 = 0.4 Find the 95% CI for p. nphat = 240 > 5, nqhat = 160 > 5, ok to use zc Step 1: Find EStep 2: Find the interval E = zc * √[pq / n] phat – E < p < phat + 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 xbar = $181,000 Normal or t-Distribution (zc or tc )? n = 18 s = $24,000 xbar = $162,000 Normal or t-Distribution?

  14. Other Topics • Finding a minimum sample size for a confidence interval • Finding zc 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 = [(zc * 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 zc for the confidence level is not provided in a table. Find the zc for an 85% CI. This zc 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 zc = - 1.44 or zc = 1.44 Note: Use the positive zc 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!

More Related