LEC21 Point & Interval Estmiates

LEC21 Point & Interval Estmiates - Point and Interval...

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Point and Interval Estimates Goal: Estimate the parameter of a distribution ex: Estimate μ from a Normal distribution ex: Estimate p from a Binomial distribution Steps: 1. Collect a “random sample” of size n 2. Compute estimate of the parameter 3. Identify the distribution of the estimate 4. Compute a confidence interval around estimate; i.e., the estimate of the mean, with 95% confidence interval is 5 +/- 1.7
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Notation z p is the point on the standard normal distribution such that the area to the right is p; i.e., P(Z z p p = ) Find z .5 z 1 z .025 2
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Confidence Interval for Estimate of μ from a Normal Distribution Assume unknown mean μ and known variance σ 2 Collect random sample: X i , i=1…n iid Compute sample statistic: X X n i i n = = 1 1 Write the distribution of the sample statistic r v X N n . . ( , ) mean = std dev = μ σ Write 95% Confidence Interval for estimate of population mean μ : X Z n ± = . . 025 196 Z .025 3
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Example A fillers fills containers with a normally distributed amount of liquid. The SD is known to be .25 ounces. A sample of 9 containers is randomly selected and the contents measured. The sample average is 32.2 ounces. Give a 95% confidence interval for the mean. Collect random sample: X i , i=1…9 iid Compute sample statistic: X X i i = = = 1 9 1 9 32 2 . Distribution of the sample statistic r v X N . . ( , ) mean = std dev =.083 32.2 95% CI for estimate of population mean μ : X Z n ± ± ± .025 163 σ 32.2 1.96(.083) 32.2 . HW p327/ 8.19d, 8.23 e 4
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90% confidence interval [ ] [ ] [ ] X z n ± = ± = ± . / . . . . . 05 32 2 165 083 32 2 137 σ 0 1.65 -1.65 0.9 .05 .05 99% confidence interval [ ] [ ] [ ] X z n ± = ± = ± . / . . . . . 005 32 2 258 083 32 2 215 5
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Two Concepts about the size of the Interval As the sample size gets larger, the interval is smaller As the confidence level gets larger, the interval is larger Z-value Interval Size 90% CI Z .05 =1.65 .137 95% CI Z .025 =1.96 .163 99% CI Z .005 =2.58 .215 6
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Interpretation of the Confidence Interval
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LEC21 Point & Interval Estmiates - Point and Interval...

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