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Confidence Intervals

Learn how to calculate a point estimate, margin of error, and interval estimate for the population mean with confidence intervals.

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Confidence Intervals

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  1. Chapter 6 Confidence Intervals

  2. Confidence Intervals for the Mean (Large Samples) § 6.1

  3.  74.22 Point Estimate for Population μ A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean, , is thesample mean, . Example: A random sample of 32 textbook prices (rounded to the nearest dollar) is taken from a local college bookstore. Find a point estimate for the population mean, . The point estimate for the population mean of textbooks in the bookstore is $74.22.

  4. Point estimate for textbooks • 74.22 interval estimate Interval Estimate An interval estimate is an interval, or range of values, used to estimate a population parameter. How confident do we want to be that the interval estimate contains the population mean, μ?

  5. c (1 – c) (1 – c) z zc z = 0 zc Critical values Level of Confidence The level of confidence c is the probability that the interval estimate contains the population parameter. c is the area beneath the normal curve between the critical values. Use the Standard Normal Table to find the corresponding z-scores. The remaining area in the tails is 1 – c .

  6. 0.90 0.05 0.05 z z = 0 zc zc Common Levels of Confidence If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean, μ. zc= 1.645 zc=  1.645 The corresponding z-scores are ± 1.645.

  7. 0.95 0.025 0.025 z z = 0 zc zc Common Levels of Confidence If the level of confidence is 95%, this means that we are 95% confident that the interval contains the population mean, μ. zc= 1.96 zc=  1.96 The corresponding z-scores are ± 1.96.

  8. 0.99 0.005 0.005 z z = 0 zc zc Common Levels of Confidence If the level of confidence is 99%, this means that we are 99% confident that the interval contains the population mean, μ. zc= 2.575 zc=  2.575 The corresponding z-scores are ± 2.575.

  9. When n 30, the sample standard deviation, s, can be used for . Margin of Error The difference between the point estimate and the actual population parameter value is called the sampling error. When μ is estimated, the sampling error is the difference μ– . Since μis usually unknown, the maximum value for the error can be calculated using the level of confidence. Given a level of confidence, the margin of error (sometimes called the maximum error of estimate or error tolerance) E is the greatest possible distance between the point estimate and the value of the parameter it is estimating.

  10. Since n 30, s can be substituted for σ. Margin of Error Example: A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample is  = 74.22, and the sample standard deviation iss = 23.44. Use a 95% confidence level and find the margin of error for the mean price of all textbooks in the bookstore. We are 95% confident that the margin of error for the population mean (all the textbooks in the bookstore) is about $8.12.

  11. Confidence Intervals for μ A c-confidence interval for the population mean μ is   E < μ< +E. The probability that the confidence interval contains μis c. Example: A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample is = 74.22, the sample standard deviation iss = 23.44, and the margin of error is E = 8.12. Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. Continued.

  12. Left endpoint = ? Right endpoint = ? • • •  = 74.22 Confidence Intervals for μ Example continued: Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. = 74.22 s = 23.44 E= 8.12   E = 74.22 – 8.12  + E = 74.22 + 8.12 = 66.1 = 82.34 With 95% confidence we can say that the cost for all textbooks in the bookstore is between $66.10 and $82.34.

  13. Finding Confidence Intervals for μ Finding a Confidence Interval for a Population Mean (n 30 or σknown with a normally distributed population) In Words In Symbols • Find the sample statistics n and . • Specify , if known. Otherwise, if n  30, find the sample standard deviation s and use it as an estimate for . • Find the critical value zc that corresponds to the given level of confidence. • Find the margin of error E. • Find the left and right endpoints and form the confidence interval. Use the Standard Normal Table. Left endpoint: ERight endpoint: +E Interval: E < μ< +E

  14. Confidence Intervals for μ( Known) Example: A random sample of 25 students had a grade point average with a mean of 2.86. Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. Construct a 90% confidence interval for the population mean grade point average. n = 25 = 2.86  =0.15 zc = 1.645 2.81 < μ< 2.91  ± E = 2.86 ± 0.05 With 90% confidence we can say that the mean grade point average for all students in the population is between 2.81 and 2.91.

  15. Sample Size Given a c-confidence level and a maximum error of estimate, E, the minimum sample size n, needed to estimate , the population mean, is If  is unknown, you can estimate it using s provided you have a preliminary sample with at least 30 members. Example: You want to estimate the mean price of all the textbooks in the college bookstore. How many books must be included in your sample if you want to be 99% confident that the sample mean is within $5 of the population mean? Continued.

  16. Sample Size Example continued: You want to estimate the mean price of all the textbooks in the college bookstore. How many books must be included in your sample if you want to be 99% confident that the sample mean is within $5 of the population mean? = 74.22   s = 23.44 zc = 2.575 (Always round up.) You should include at least 146 books in your sample.

  17. Confidence Intervals for the Mean (Small Samples) § 6.2

  18. The t-Distribution When a sample size is less than 30, and the random variable x is approximately normally distributed, it follow a t-distribution. Properties of the t-distribution • The t-distribution is bell shaped and symmetric about the mean. • The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as  is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size. • d.f. = n – 1 Degrees of freedom Continued.

  19. d.f. = 2 d.f. = 5 t Standard normal curve 0 The t-Distribution • The total area under a t-curve is 1 or 100%. • The mean, median, and mode of the t-distribution are equal to zero. • As the degrees of freedom increase, the t-distribution approaches the normal distribution. After 30 d.f., the t-distribution is very close to the standard normal z-distribution. The tails in the t-distribution are “thicker” than those in the standard normal distribution.

  20. Critical Values of t Example: Find the critical value tc for a 95% confidence when the sample size is 5. Appendix B: Table 5: t-Distribution d.f. = n – 1 = 5 – 1 = 4 tc = 2.776 c = 0.95 Continued.

  21. c = 0.95 t tc =  2.776 tc = 2.776 Critical Values of t Example continued: Find the critical value tc for a 95% confidence when the sample size is 5. 95% of the area under the t-distribution curve with 4 degrees of freedom lies between t = ±2.776.

  22. Confidence Intervals and t-Distributions Constructing a Confidence Interval for the Mean: t-Distribution In Words In Symbols • Identify the sample statistics n, , and s. • Identify the degrees of freedom, the level of confidence c, and the critical value tc. • Find the margin of error E. • Find the left and right endpoints and form the confidence interval. d.f. = n – 1 Left endpoint: ERight endpoint: +E Interval: E < μ< +E

  23. Constructing a Confidence Interval Example: In a random sample of 20 customers at a local fast food restaurant, the mean waiting time to order is 95 seconds, and the standard deviation is 21 seconds. Assume the wait times are normally distributed and construct a 90% confidence interval for the mean wait time of all customers. n = 20 = 95 s = 21 d.f.= 19 tc = 1.729 86.9 < μ< 103.1  ± E = 95 ± 8.1 We are 90% confident that the mean wait time for all customers is between 86.9 and 103.1 seconds.

  24. Use the normal distribution with If  is unknown, use s instead. Yes Yes No No Yes Use the normal distribution with No Use the t-distribution with and n – 1 degrees of freedom. Normal or t-Distribution? Is n  30? Is the population normally, or approximately normally, distributed? You cannot use the normal distribution or the t-distribution. Is  known?

  25. Normal or t-Distribution? Example: Determine whether to use the normal distribution, the t-distribution, or neither. a.) n = 50, the distribution is skewed, s = 2.5 The normal distribution would be used because the sample size is 50. b.) n = 25, the distribution is skewed, s = 52.9 Neither distribution would be used because n < 30 and the distribution is skewed. c.) n = 25, the distribution is normal,  = 4.12 The normal distribution would be used because although n < 30, the population standard deviation is known.

  26. Confidence Intervals for Population Proportions § 6.3

  27. Point Estimate for Population p The probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by where x is the number of successes in the sample and n is the number in the sample. The point estimate for the proportion of failures is = 1 – The symbols and are read as “p hat” and “q hat.”

  28. Point Estimate for Population p Example: In a survey of 1250 US adults, 450 of them said that their favorite sport to watch is baseball. Find a point estimate for the population proportion of US adults who say their favorite sport to watch is baseball. n = 1250 x = 450 The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%.

  29. Confidence Intervals for p A c-confidence interval for the population proportion p is where The probability that the confidence interval contains p is c. Example: Construct a 90% confidence interval for the proportion of US adults who say baseball is their favorite sport to watch. n = 1250 x = 450 Continued.

  30. Left endpoint = ? Right endpoint = ? • • • Confidence Intervals for p Example continued: n = 1250 x = 450 With 90% confidence we can say that the proportion of all US adults who say baseball is their favorite sport to watch is between 33.8% and 38.2%.

  31. Left endpoint: Right endpoint: Interval: Finding Confidence Intervals for p Constructing a Confidence Interval for a Population Proportion In Words In Symbols • Identify the sample statistics n and x. • Find the point estimate • Verify that the sampling distribution can be approximated by the normal distribution. • Find the critical value zc that corresponds to the given level of confidence. • Find the margin of error E. • Find the left and right endpoints and form the confidence interval. Use the Standard Normal Table.

  32. Sample Size Given a c-confidence level and a margin of error, E, the minimum sample size n, needed to estimate p is This formula assumes you have an estimate for and If not, use and Example: You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. Continued.

  33. Sample Size Example continued: You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. n = 1250 x = 450 (Always round up.) You should sample at least 2213 adults to be 95% confident.

  34. Confidence Intervals for Variance and Standard Deviation § 6.4

  35. The Chi-Square Distribution The pointestimate for 2 is s2, and the point estimate for is s. s2 is the most unbiased estimate for 2. You can use the chi-square distribution to construct a confidence interval for the variance and standard deviation. If the random variable x has a normal distribution, then the distribution of forms a chi-square distribution for samples of any size n > 1.

  36. The Chi-Square Distribution Four properties of the chi-square distribution are as follows. • All chi-square values χ2 are greater than or equal to zero. • The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for 2, use the χ2-distribution with degrees of freedom. To form a confidence interval for 2, use the χ2-distribution with degrees of freedom equal to one less than the sample size. • The area under each curve of the chi-square distribution equals one. • Find the critical value zc that corresponds to the given level of confidence. • Chi-square distributions are positively skewed.

  37. X2 X2 X2R X2L Area to the right of X2R Area to the right of X2L c X2 X2R X2L Critical Values for X2 There are two critical values for each level of confidence. The value χ2R represents the right-tail critical value and χ2L represents the left-tail critical value. The area between the left and right critical values is c.

  38. Area to the right of χ2R = Area to the right of χ2L = Critical Values for X2 Example: Find the critical values χ2R and χ2L for an 80% confidence when the sample size is 18. Because the sample size is 18, there are d.f. = n– 1 = 18 – 1 = 17 degrees of freedom, Use the Chi-square distribution table to find the critical values. Continued.

  39. Critical Values for X2 Example continued: Appendix B: Table 6: χ2-Distribution χ2R = 24.769 χ2L = 10.085

  40. Confidence Intervals for 2 and  A c-confidence interval for a population variance and standard deviation is as follows. Confidence Interval for 2: Confidence Interval for : The probability that the confidence intervals contain 2 or  is c.

  41. Confidence Intervals for 2and  Constructing a Confidence Interval for a Variance and a Standard Deviation In Words In Symbols • Verify that the population has a normal distribution. • Identify the sample statistic n and the degrees of freedom. • Find the point estimate s2. • Find the critical value χ2R and χ2L that correspond to the given level of confidence. Use Table 6 in Appendix B. Continued.

  42. Confidence Intervals for 2and  Constructing a Confidence Interval for a Variance and a Standard Deviation In Words In Symbols • Find the left and right endpoints and form the confidence interval. • Find the confidence interval for the population standard deviation by taking the square root of each endpoint.

  43. Area to the right of χ2R = Area to the right of χ2L = Constructing a Confidence Interval Example: You randomly select and weigh 41 samples of 16-ounce bags of potato chips. The sample standard deviation is 0.05 ounce. Assuming the weights are normally distributed, construct a 90% confidence interval for the population standard deviation. d.f. = n– 1 = 41 – 1 = 40 degrees of freedom, The critical values are χ2R = 55.758 and χ2L = 26.509. Continued.

  44. Left endpoint = ? Right endpoint = ? • • Constructing a Confidence Interval Example continued: χ2L = 26.509 χ2R = 55.758 With 90% confidence we can say that the population standard deviation is between 0.04 and 0.06 ounces.

  45. Homework • Page 321 • 2,4,7,12,16,20,23,31,42,45

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