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Chapter 7

Confidence Intervals. Chapter 7 . Order of this Chapter. We are going to do this chapter a little out of order: 7.4: Proportions 7.2: Means (with known variance or sample size greater than 30) 7.3: Means (with unknown variance and sample size smaller than 30)

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Chapter 7

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  1. Confidence Intervals Chapter 7

  2. Order of this Chapter • We are going to do this chapter a little out of order: • 7.4: Proportions • 7.2: Means (with known variance or sample size greater than 30) • 7.3: Means (with unknown variance and sample size smaller than 30) • 7.5: Variances and Standard Deviations

  3. Review • Parameter (Population Characteristics) • m (mean) • p (proportion) • Statistic (Sample Characteristics) • (sample mean) • (sample proportion)

  4. Review • “Statistics will be different for each sample. These differences obey certain laws of probability (but only for random samples).” • Taking a sample from a population is a random phenomena. That means: • The outcome is unknown before the event occurs • The long term behaviour is predictable

  5. Sampling Distribution Models PopulationParameter? Population Population Inference Sample Statistic Sample

  6. Example • 2002 Senators Population Characteristics • p = proportion of Democratic Senators • μ = mean number of years in Senate • Take SRS of size n = 10 • Calculate Sample Characteristics • = sample proportion of Democratic Senators • = sample mean number of years in Senate

  7. Sample 1 0.5 15.1 2 0.5 15.0 3 0.7 13.5 4 0.6 11.0 5 0.4 16.3 Example

  8. SRS characteristics • Values of and are random • Change from sample to sample • Different from population characteristics • p = 0.50 • μ = 12.38 years

  9. Imagine • Repeat taking SRS of size n = 10 • Collection of values for and • Collection of and values ARE DATA • Summarize data – make a histogram • Shape, Center and Spread • Sampling distribution for and (when we know σ or when n ≥ 30) is Normal!!

  10. n=2 n=5 n=10 n=25 Example • 50% of people on campus favor current academic calendar. • Select n people. • Find sample proportion of people favoring current academic calendar. • Repeat sampling. • What does sampling distribution of sample proportion look like?

  11. Election 2004 • You have seen Confidence Intervals before! • Oct. 9-10 CNN/USA Today/Gallup Poll (793 likely voters) • Kerry 49% • President Bush 48% • Margin of Error: ± 3%

  12. Confidence Intervals • Confidence intervals come from the fact that we could take multiple samples and calculate multiple 95% confidence intervals and, if we were using the same method to find all the intervals, we would expect that about 95% of the intervals we constructed would contain the true parameter (population proportion).

  13. Construction of CI • , the point estimate, is the center of the interval. It merely shifts the interval along the axis. • , the critical value, is the number of multiples of the standard error needed to form the desired CI. This is will depend on the level of confidence you want.

  14. How to find • Need z-tables • Based on Normal model • Between what two z-values do 95% of the observations lie on N(0,1)?

  15. Most Common z*

  16. Confidence Interval for p • So, the CI for p looks like

  17. Interpret the interval • I am ___% confident that the true population proportion falls within the interval I constructed.

  18. Example 1 CI for p • A survey of 200,000 boat owners found that 12% of the pleasure boats were named Serenity. Find the 95% confidence interval of the true proportion of boats named Serenity.

  19. Example 1 CI for p (cont.) • Construct the interval

  20. Example 1 CI for p (cont.) • So our interval is (0.119, 0.121) • Interpretation: • “I am 95% confident that the true proportion of boats named Serenity is between 11.9% and 12.1%”

  21. Sample Size and the ME • Determining sample size. • We know that ME = z* • We can manipulate this equation with algebra.

  22. Example 2 Sample Size for p • A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary.

  23. Example 2 Sample Size for p (cont.) • Formula: • n = 2304.96 ≈ 2305 people to interview

  24. Confidence Interval for μ(when we know the variance or n ≥ 30) • So, the CI for μ looks like

  25. Interpret the interval • I am ___% confident that the true population mean falls within the interval I constructed.

  26. Example 3 CI for μ(when we know the variance or n ≥ 30) • Noise levels at various area urban hospitals were measured in decibels. The mean of he noise levels in 84 corridors was 61.2 decibels, and the standard deviation was 7.9. Find the 95% confidence interval of the true mean.

  27. Example 3 CI for μ(when we know the variance or n ≥ 30) • Construct the interval

  28. Example 3 CI for μ(when we know the variance or n ≥ 30) • So our interval is (59.5, 62.9) • Interpretation: • “I am 95% confident that the true mean corridor noise level is between 59.5 and 62.9 decibels.”

  29. Sample Size for μ • Formula:

  30. Example 4 Sample Size for μ • A university dean of students wishes to estimate the average number of hours students spend doing homework per week. The standard deviation from a previous study is 6.2 hours. How large a sample must be selected if he wants to be 99% confident of finding whether the true mean differs from the sample mean by 1.5 hours?

  31. Example 4 Sample Size for μ • Formula:

  32. Confidence Interval for μ(when we do not know the variance and n < 30) • So, the CI for μ looks like:

  33. Example 5 CI for μ(when we do not know the variance and n < 30) • A sample of six college wrestlers had an average weight of 276 pounds with a sample standard deviation of 12 pounds. Find the 90% confidence interval of the true mean weight of all college wrestlers.

  34. Example 5 CI for μ(when we do not know the variance and n < 30) • Construct the interval

  35. Example 5 CI for μ(when we do not know the variance and n < 30) • So our interval is (266.13, 285.87) • Interpretation: • “I am 95% confident that the true mean weight of college wrestlers is between 266.13 and 285.87 pounds.”

  36. Confidence Interval for σ2 • So, the CI for σ2 looks like:

  37. Example 6 CI for σ2 • A service station advertises that customers will have to wait no more than 30 minutes for an oil change. A sample of 28 oil changes has a standard deviation of 5.2 minutes. Find the 95% confidence interval of the population standard deviation of the time spent waiting for an oil change.

  38. Example 6 CI for σ2 • To use the chi-squared table you need to find α/2 and (1 – α/2). • To find α take (1 – confidence level) • Then divide by 2. This is α/2.

  39. Example 6 CI for σ2 • Construct the interval

  40. Example 6 CI for σ2 • So our interval is (4.11, 7.08) • Interpretation: • “I am 95% confident that the true standard deviation of customers who have to wait no more than 30 minutes for an oil change is between 4.11 and 7.08 minutes.”

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