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Computing and Interpreting Appropriate Confidence Intervals Prof Catherine Comiskey

Computing and Interpreting Appropriate Confidence Intervals Prof Catherine Comiskey. Overview. Computing a confidence interval (CI) for a sample mean (average) when the population standard deviation (sigma) is known

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Computing and Interpreting Appropriate Confidence Intervals Prof Catherine Comiskey

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  1. Computing and Interpreting Appropriate Confidence IntervalsProf Catherine Comiskey School of Nursing and Midwifery 24 D'Olier Street Dublin 2

  2. Overview • Computing a confidence interval (CI) for a sample mean (average) when the population standard deviation (sigma) is known • Using the t distribution and computing CI for a sample mean when the population standard deviation is unknown • Computing a CI for a sample proportion. School of Nursing and Midwifery 24 D'Olier Street Dublin 2

  3. Confidence Intervals and Hypothesis Tests • Estimating Confidence Intervals, CI • Means and Proportions • Examples and Exercises

  4. Margin of Error and the Interval Estimate A point estimator cannot be expected to provide the exact value of the population parameter. An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. Point Estimate +/- Margin of Error The purpose of an interval estimate is to provide information about how close the point estimate is to the value of the parameter.

  5. Margin of Error and the Interval Estimate The general form of an interval estimate of a population mean is

  6. Interval Estimation of a Population Mean:s Known • In order to develop an interval estimate of a population mean, the margin of error must be computed using either: • the population standard deviation s , or • the sample standard deviation s • s is rarely known exactly, but often a good estimate can be obtained based on historical data or other information. • We refer to such cases as the s known case.

  7. where: is the sample mean 1 - is the confidence coefficient z/2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution s is the population standard deviation n is the sample size Interval Estimate of a Population Mean:s Known (key slide) • Interval Estimate of m

  8. Interval Estimate of a Population Mean:s Known • Adequate Sample Size In most applications, a sample size of n = 30 is adequate. If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended.

  9. Interval Estimate of a Population Mean:s Known • Adequate Sample Size (continued) If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice. If the population is believed to be at least approximately normal, a sample size of less than 15 can be used.

  10. D S Interval Estimate of Population Mean: Known • Example: Discount Sounds Discount Sounds has 260 retail outlets throughout the country. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. A sample of size n = 36 was taken; the sample mean income was €31,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be €4,500, and the confidence coefficient to be used in the interval estimate is .95.

  11. D S 95% of the sample means that can be observed are within + 1.96 of the population mean . Interval Estimate of Population Mean: Known (key slide) The margin of error is: Thus, at 95% confidence, the margin of error is €1,470.

  12. D S Interval Estimate of Population Mean: Known (key slide) Interval estimate of  is: €31,100 + €1,470 or €29,630 to €32,570 We are 95% confident that the interval contains the population mean.

  13. Class Exercise 1 During a work health and safety week a random sample of 55 employees had their blood pressure recorded. Results in mm of Hg gave a sample mean 102.4. It is known that individuals with the same age distribution as the employees have a blood pressure standard deviation of 10.5mm. Compute a 95% CI for the mean blood pressure of the population of employees.

  14. Interval Estimation of a Population Mean:s Unknown • If an estimate of the population standard deviation s cannot be developed prior to sampling, we use the sample standard deviation s to estimate s . • This is the s unknown case. • In this case, the interval estimate for m is based on the t distribution. • (We’ll assume for now that the population is normally distributed.)

  15. t Distribution The t distribution is a family of similar probability distributions. A specific t distribution depends on a parameter known as the degrees of freedom. Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.

  16. t Distribution A t distribution with more degrees of freedom has less dispersion. As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.

  17. t Distribution t distribution (20 degrees of freedom) Standard normal distribution t distribution (10 degrees of freedom) z, t 0

  18. The standard normal z values can be found in the infinite degrees () row of the t distribution table. t Distribution For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value.

  19. t Distribution Standard normal z values

  20. Interval Estimation of a Population Mean:s Unknown (key slide) • Interval Estimate where: 1 - = the confidence coefficient t/2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation

  21. Interval Estimation of a Population Mean:s Unknown A reporter for a regional newspaper is writing an article on the cost of family housing. A sample of 16 apartments within a half-mile of the town centre resulted in a sample mean of €650 per month and a sample standard deviation of €55. • Example: Family Rents

  22. Interval Estimation of a Population Mean:s Unknown • Example: Family Rents Let us provide a 95% confidence interval estimate of the mean rent per month for the population of apartments within a half-mile of town. We will assume this population to be normally distributed.

  23. Interval Estimation of a Population Mean:s Unknown At 95% confidence,  = .05, and /2 = .025. t.025 is based on n- 1 = 16 - 1 = 15 degrees of freedom. In the t distribution table we see that t.025 = 2.131.

  24. Interval Estimation of a Population Mean:s Unknown (key slide) • Interval Estimate Margin of Error We are 95% confident that the mean rent per month for the population of family apartments within a half-mile of town is between €620.70 and €679.30.

  25. Summary of Interval Estimation Procedures for a Population Mean Can the population standard deviation s be assumed known ? Yes No Use the sample standard deviation s to estimate s s Known Case Use Use s Unknown Case

  26. Interval Estimationof a Population Proportion The general form of an interval estimate of a population proportion is

  27. The sampling distribution of plays a key role in computing the margin of error for this interval estimate. The sampling distribution of can be approximated by a normal distribution whenever np> 5 and n(1 – p) > 5. Interval Estimationof a Population Proportion

  28. where: 1 - is the confidence coefficient z/2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution is the sample proportion Interval Estimationof a Population Proportion • Interval Estimate

  29. Example • Barron, Comiskey and Saris (2009) BMI in 900 Irish children

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