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Confidence Intervals

Confidence Intervals. Mon, March 22 nd. Point & Interval Estimates. Point estimate – use sample to estimate exact statistic to represent pop parameter Point estimate of average Amer salary = $29, 340

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Confidence Intervals

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  1. Confidence Intervals Mon, March 22nd

  2. Point & Interval Estimates • Point estimate – use sample to estimate exact statistic to represent pop parameter • Point estimate of average Amer salary = $29, 340 • Interval estimate – use sample to estimate a range of values within which pop parameter may fall (Confidence Interval) • Interval estimate of average salary = $27,869 to $30,811

  3. (cont.) • With confidence interval, specify likelihood this interval will contain the pop parameter • 95% conf interval, means we are 95% confident the interval/range contains the true pop parameter • Almost always choose 90, 95, or 99% confidence

  4. Constructing Confidence Interval • 1) Calculate standard error of the mean ybar = y / sqrt N • 2) Decide on confidence level (90/95/99) – then find corresponding z value • We know that, for a normal curve, 68% of the scores will fall betw + or – 1SD (std error), so • 95% will fall betw + or – 1.96 SE (see normal curve table for .05 / 2 tails, so z = + or –1.96 • 99% will fall betw + or –2.58 SE (see normal curve table for .01 / 2 tails, so z = + or – 2.58)

  5. (cont.) • 3) Use Conf Interval formula: CI = Ybar + and – Z(ybar ) • 4) Interpret results Ex) Find 95% CI for average commuting time when ybar = 7.5 hrs, y = 1.5 and sample N=500 *Find standard error, ybar = 1.5 / sqrt(500) = .07

  6. example • For 95% CI, z value is 1.96 (see table 12.1 for z values for 90/95/99% CI) • 95% CI = 7.5 + and – 1.96(.07) = 7.36 to 7.64 • (7.36, 7.64) • Interpretation – we are 95% confident the true commuting time of the pop is between 7.36 and 7.64 hrs per week)

  7. Example (cont.) • Notice what happens to CI when we increase confidence to 99% • Corresponding z for 99% = 2.58, so • 99% CI = 7.5 + and – 2.58(.07) = 7.32 to 7.68 • Now only 1% risk we are wrong, but a wider, less precise, interval

  8. Estimating ybar • If not given y and only given Sy (sample std dev), can estimate Sybar (rather than ybar) • Sybar = Sy / sqrt N

  9. Sample Size and CI • Increase N and increase precision of CI (range becomes smaller): • Due to smaller standard error • Earlier example, increase N from 500 to 2500, ybar = 1.5 / sqrt(500) = .07 ybar = 1.5 / sqrt(2500) = .03 CI = 7.5 + and – 1.96(.03) = 7.44 to 7.51 Compared w/7.36 to 7.64 (w/ .07 std error)

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