Confidence Intervals Chapter 6
§ 6.1 Confidence Intervals for the Mean (Large Samples)
Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 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 the sample 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
Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Interval Estimate An interval estimate is an interval, or range of values, used to estimate a population parameter. Point estimate for textbooks interval estimate How confident do we want to be that the interval estimate contains the population mean, μ?
Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Level of Confidence The level of confidence c is the probability that the interval estimate contains the population parameter. z z = 0 zczc zczc Critical values (1 – c) c is the area beneath the normal curve between the critical values. The remaining area in the tails is 1 – c. c Use the Standard Normal Table to find the corresponding z - scores.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 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. z z = 0 zczc zczc The corresponding z - scores are ± z c = z c = 1.645
Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 z z = 0 zczc zczc 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. The corresponding z - scores are ± z c = 1.96 z c = 1.96
Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 z z = 0 zczc zczc 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. The corresponding z - scores are ± z c = z c = 2.575
Larson & Farber, Elementary Statistics: Picturing the World, 3e 9 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 μ is 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. When n 30, the sample standard deviation, s, can be used for .
Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 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 is s = 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. Since n 30, s can be substituted for σ.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 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 μ is 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 is s = 23.44, and the margin of error is E = Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 12 Confidence Intervals for μ Example continued: Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. = 74.22s = 23.44E = 8.12 With 95% confidence we can say that the cost for all textbooks in the bookstore is between $66.10 and $ = Left endpoint = ?Right endpoint = ? E = – E = = 66.1= 82.34
Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Finding Confidence Intervals for μ Finding a Confidence Interval for a Population Mean (n 30 or σ known with a normally distributed population) In Words In Symbols 1.Find the sample statistics n and. 2.Specify , if known. Otherwise, if n 30, find the sample standard deviation s and use it as an estimate for . 3.Find the critical value z c that corresponds to the given level of confidence. 4.Find the margin of error E. 5.Find the left and right endpoints and form the confidence interval. Use the Standard Normal Table. Left endpoint: E Right endpoint: +E Interval: E < μ < +E
Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 Confidence Intervals for μ ( Known) Example: A random sample of 25 students had a grade point average with a mean of 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 z c = ± E = 2.86 ± < μ < 2.91 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.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 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.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 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? = s = z c = You should include at least 146 books in your sample. (Always round up.)