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6.1 Confidence intervals for the mean (large samples)

6.1 Confidence intervals for the mean (large samples). Find a point estimate and a margin of error Construct and interpret confidence intervals for the population mean Determine minimum sample size required when estimating µ.

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6.1 Confidence intervals for the mean (large samples)

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  1. 6.1 Confidence intervals for the mean (large samples) Find a point estimate and a margin of error Construct and interpret confidence intervals for the population mean Determine minimum sample size required when estimating µ

  2. A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean µ is the sample mean. Point estimate

  3. Try it yourself 1 • Finding a Point Estimate A social networking website allows its users to add friends, send messages, and update their personal profiles. The following represents a random sample of the number of friends for 30 users of the website. Find a point estimate of the population mean µ. 138.5

  4. An interval estimate is an interval, or range of values, used to estimate a population parameter. Interval estimate

  5. The level of confidence c is the probability that the interval estimate contains the population parameter. Level of confidence c

  6. Given a level of confidence c, the margin of error E (sometimes also called the maximum error of estimate or error tolerance) is the greatest possible distance between the point estimate and the value of the parameter it is estimating. In order to use this technique, it is assumed that the population standard deviation is known. This is rarely the case, but when n > 30, the sample standard deviation s can be used in place of σ. Margin of error e

  7. Try it yourself 2 • Finding the Margin of Error Use the data given in Try It Yourself 1 and a 95% confidence interval level to find the margin of error for the mean number of friends for all users of the website. E = 18.3

  8. A c-confidence interval for the population mean µ is The probability that the confidence interval contains µ is c. C-confidence interval for the population mean µ

  9. Try it yourself 3 • Constructing a Confidence Interval Use the data given in Try It Yourself 1 to construct 95% confidence interval for the mean number of friends for all users of the website. 95% CI (120.2,156.8)

  10. Try it yourself 4 • Constructing a Confidence Interval Using Technology Use the sample data in Example 1 and a technology tool to construct 75%, 85%, and 99% confidence intervals for the mean number of friends for all users of the website. How does the width of the confidence interval change as the level of confidence increases.

  11. Try it yourself 4 • Constructing a Confidence Interval Using Technology 75% CI (121.2, 140.4) 85% CI (118.7, 142.9) 99% CI (109.2, 152.4)

  12. Try it yourself 5 • Constructing a Confidence Interval, σ Known A college admissions director wishes to estimate the mean age of all students currently enrolled. In a random sample of 30 students, the mean age is found to be 22.9 years. From past studies, the standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval of the population mean age. 90% CI (22.4, 23.4)

  13. Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean µ is If σ is unknown, you can estimate it using s, provided you have a preliminary sample with at least 30 members. Find a minimum sample size to estimate µ

  14. Try it yourself 6 • Determining a Minimum Sample Size You want to estimate the mean number of friends for all users of the website. How many users must be included in the sample if you want to be 95% confident that the sample mean is within 10 users of the population mean? n =

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