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Understanding Confidence Intervals in Estimations

Learn how to estimate population parameters with confidence intervals, including interpreting point estimates, margins of error, and critical values. Understand confidence levels and sample sizes in statistical analysis.

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Understanding Confidence Intervals in Estimations

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  1. Chapter 7Confidence Intervals and Sample Sizes 7.2 Estimating a Proportionp 7.3 Estimating a Meanµ (σknown) 7.4 Estimating a Meanµ (σunknown) 7.5 Estimating a Standard Deviationσ

  2. Example 1 In a recent poll, 70% or 1501 randomly selected adults said they believed in global warming. Q: What is the proportion of the adult population that believe in global warming? TRICK QUESTION! We only know the sample proportion s, We do not know the population proportion σ. BUT… The proportion of the sample (0.7) is our best point estimate (i.e. best guess).

  3. Definition Point Estimate A single value (or point) used to approximate a population parameter Best Point Estimate PopulationParameter p µ σ ≈ ≈ ≈ p x s Proportion Mean Std. Dev.

  4. Definition Confidence Interval : CI The range (or interval) of values to estimate the true value of a population parameter. It is abbreviated as CI In Example 1, the 95% confidence interval for the population proportion p is CI = (0.677, 0.723)

  5. Definition Confidence Level : 1 – α The probability that the confidence interval actually contains the population parameter. The most common confidence levels used are 90%, 95%, 99% 90% : α=0.1 95% : α=0.05 99% : α=0.01 In Example 1, the Confidence level is 95%

  6. Definition Margin of Error : E The maximum likely difference between the observed value and true value of the population parameter (with probability is 1–α) The margin of error is used to determine a confidence interval (of a proportion or mean) In Example 1, the 95% margin of error for the population proportion p is E = 0.023

  7. Continued… Example 1 In a recent poll, 70% or 1501 randomly selected adults said they believed in global warming. Q: What is the proportion of the adultpopulation that believe in global warming? A:0.7is the best point estimate of the proportion of all adults who believe in global warming. The 95% confidence interval of the population proportion p is: CI = (0.677, 0.723) ( with a margin of error E = 0.023 ) What does it mean, exactly?

  8. Interpreting a Confidence Interval For the 95% confidence interval CI = (0.677, 0.723)we say: We are 95% confident that the interval from 0.677 to 0.723 actually does contain the true value of the population proportion p. This means that if we were to select many different samples of size 1501 and construct the corresponding confidence intervals, then 95% of them would actually contain the value of the population proportion p.

  9. !!! Caution !!! Know the correct interpretation of a confidence interval It is incorrectto say “ the probability that the population parameter belongs to the confidence interval is 95% ” because the population parameter is not a random variable, its value cannot change The population is “set in stone”

  10. !!! Caution !!! Do not confuse the two percentages The proportioncan be represented by percents (like 70% in Example 1) The confidence level may be represented by percents (like 95% in Example 1) Proportions can be any value from 0% to 100% Confidence levels are usually 90%, 95%, or 99%

  11. Confidence Interval Formula ( y – E, y + E ) y= Best point estimate E= Margin of Error • Centered at the best point estimate • Width is determined by E • The value of E depends the critical valueof the CI

  12. Finding the Point Estimate and E from a Confidence Interval Point estimate : y y=(upper confidence limit) + (lower confidence limit) 2 Margin of Error : E E = (upper confidence limit) — (lower confidence limit) 2

  13. Definition Critical Value The number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. A critical value is dependent on a probability distribution the parameter follows and the confidence level (1 – α) .

  14. Normal Dist. Critical Values For a population proportionp and meanµ(σ known), the critical values are found using z-scores on a standard normal distribution The standard normal distribution is divided into three regions: middle part has area 1 – α and two tails (left and right) have area α/2 each:

  15. Normal Dist. Critical Values The z-scores za/2and–za/2separate the values: Likely values ( middle interval ) Unlikely values ( tails ) Use StatCrunch to calculate z-scores (see Ch. 6) za/2 –za/2

  16. Normal Dist. Critical Values The valuez/2separates an area of/2in the right tail of the z-dist. The value –z/2 separates an area of/2 in the left tail of the z-dist. The subscript/2issimply a reminder that thez-score separates an areaof /2in the tail.

  17. Section 7.2Estimating a Population Proportion Objective Find the confidence intervalfor a population proportion p Determine the sample sizeneeded to estimate a population proportion p

  18. Definitions The best point estimatefor a population proportion p is the sample proportion p Best point estimate :p The margin of error Eis the maximum likely difference between the observed value and true value of the population proportion p (with probability is 1–α)

  19. Margin of Error for Proportions E= margin of error p= sample proportion q= 1 – p n= number sample values 1 – α = Confidence Level

  20. ˆ ˆ ( p – E, p + E ) Confidence Interval for a Population Proportion p where

  21. Finding the Point Estimate and E from a Confidence Interval Point estimate of p: p=(upper confidence limit) + (lower confidence limit) 2 Margin of Error: E = (upper confidence limit) — (lower confidence limit) 2

  22. Round-Off Rule for Confidence Interval Estimates of p • Round the confidence interval limits for p to three significant digits.

  23. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Direct Computation

  24. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Using StatCrunch Stat → Proportions → One Sample → with Summary

  25. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Using StatCrunch Enter Values

  26. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Using StatCrunch Click ‘Next’

  27. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Using StatCrunch Select ‘Confidence Interval’

  28. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Using StatCrunch Enter Confidence Level, then click ‘Calculate’

  29. Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Using StatCrunch Standard Error Lower Limit Upper Limit From the output, we find the Confidence interval is CI = (0.578, 0.762)

  30. Sample Size Suppose we want to collect sample data in order to estimate some population proportion. The question is how manysample items must be obtained?

  31. ˆ ˆ p q z E= a / 2 n (solve for n by algebra) ˆ ˆ Z ()2 a / 2 p q n = E 2 Determining Sample Size

  32. ˆ When an estimate of p is known: ˆ z ˆ ()2 p q a / 2 n = E 2 ˆ When no estimate of p is known: use p = q = 0.5 z ()2 a / 2 0.25 n = E 2 Sample Size for Estimating Proportion p ˆ ˆ

  33. Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of nup to the next larger whole number. Examples: n = 310.67 round up to 311 n = 295.23 round up to 296 n = 113.01 round up to 114

  34. Example 2 A manager for E-Bay wants to determine thecurrent percentage of U.S. adults who now use the Internet. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points when… (a) In 2006, 73% of adults used the Internet. (b) No known possible value of the proportion.

  35. Example 2 (a) Given: Given a sample has proportion of 0.73, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 842 adults.

  36. Example 2 (b) Given: For any sample, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 1068 adults.

  37. SummaryConfidence Interval of a Proportion E= margin of error p= sample proportion n= number sample values 1 – α = Confidence Level ( p – E, p + E )

  38. ˆ z ˆ ()2 p q a / 2 n = E 2 When no estimate of p is known (use p = q = 0.5) z ()2 a / 2 0.25 n = E 2 SummarySample Size for Estimating a Proportion When an estimate of p is known:

  39. Section 7.3Estimating a Population mean µ(σ known) Objective Find the confidence intervalfor a population mean µ when σ is known Determine the sample size needed to estimate a population mean µ when σ is known

  40. Best Point Estimation The best point estimatefor a population mean µ (σ known) is the sample mean x Best point estimate :x

  41. Notation = population mean = population standard deviation = sample mean n = number of sample values E= margin of error z/2 = z-score separating an area of α/2 in the right tail of the standard normal distribution

  42. Requirements (1) The population standard deviation σis known (2) One or both of the following: The population is normally distributed or n > 30

  43. Margin of Error

  44. Confidence Interval ( x – E, x + E ) where

  45. Definition The two values x – E and x + E are called confidence interval limits.

  46. Round-Off Rules for Confidence Intervals Used to Estimate µ • When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data. • When the original set of data is unknown and only the summary statistics(n, x, s)are used, round the confidence interval limits to the same number of decimal places used for the sample mean.

  47. Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Direct Computation

  48. Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Stat → Z statistics → One Sample → with Summary

  49. Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Enter Parameters

  50. Example Find the 90% confidence interval for the population mean If the population standard deviation is known to be 10 and a sample of size 42 has a mean of 38.4 Using StatCrunch Click Next

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