251paraml 06/28/06 (Open this document in 'Outline' view!)O. Estimation of Parameters.1. Point and Interval Estimation.Properties of Estimators. Let be an estimator for .a. Unbiassedness .b. Consistency (As sample size gets larger, estimate gets better).c. Efficiency ( has a small variance). Define BLUE.d. Maximum Likelihood ( is the value of that is mostlikely to have produced the observed data) .2. A Confidence Interval for When is Known.Step 1: State the confidence level and significance level.The given confidence level of 95% represents the probability thatthe interval actually contains the mean and is stated asThe significance level of 5% represents the probability of beingwrong and isStep 2: Find the appropriate value of Use the last line ofTable 18 (or Table 17) in the Syllabus Supplement to find(the bottom number in the .025 column).Note that higher confidence levels give larger values of , andthus larger confidence intervals.Step 3: Find the standard error.Note that larger values of make the standard error and theconfidence interval smaller.Step 4: Put it together.. The last part of this expression means that theinterval extends from 62 – 23.52 = 38.48 to 62 + 23.52 = 85.52.The result can be writtenExample 2: Assume that a population is Normally distributed withStep 1: State the confidence level and significance level.The given confidence level of 99% represents the probability thatthe interval actually contains the mean and is stated asThe significance level of 1% represents the probability of beingwrong and isStep 2: Find the appropriate value of Use the last line ofTable 18 (or Table 17) in the Syllabus Supplement to find(the bottom number in the .005 column).Note that higher confidence levels give larger values of ,and thus larger confidence intervals.Step 3: Find the standard error.No change from example 1.Step 4: Put it together..The result can be writtenOr make a Normal curve with 62 in the middle and 31.09 and 92.91on the sides. Label the area between 31.09 and 92.91 with 99%,the area below 31.09 with 0.5% and the area above 92.91 with0.5%.Definitions:3. A Confidence Interval for When is not known.Step1: State the confidence level and significance level.The given confidence level of 95% represents the probability thatthe interval actually contains the mean and is stated asThe significance level of 5% represents the probability of beingwrong and isStep2: Find the appropriate value of Use Table 18 in theSyllabus Supplement to find (the number inthe .025 column and the 9th row). Note that higher confidencelevels and lower numbers of degrees of freedom give larger valuesof , and thus larger confidence intervals.Step 3: Find the standard error.Step 4: Put it together. .The result can be writtenOr make a ‘Normal’ curve with 838 in the middle and 759.3 and916.7 on the sides. Label the area between 759.3 and 916.7 with95%, the area below 759.3 with 2.5% and the area above 916.7with 2.5%.Example 2: Find a 98% confidence interval for the mean whenandStep1: Confidence level is 98%, so that the significance level isStep 2: Since is a sample std. deviation,Step 3:Step 4: .You should express this as an interval.Example 3: We visit a town of 5000 families. We take a sample of900 families and find a sample mean of $8536 and a sample standarddeviation of $436.Find a 90% confidence interval for the mean.andStep1: Confidence level is 90%, so that the significance level isStep 2: Since the degrees of freedom are we run offthe table. If the degrees of freedom are much over 200, use thevalue from the infinity line.Step 3: This is the big change. Since the sample is more than 5%of the population, use the finite population correction.Note that the smaller the population, the more the finite populationcorrection will shrink the standard error.Step 4: .You should express this as an interval.4. A Confidence Interval for a Proportion©2002 Roger Even Bove251paraml 06/28/06 (Open this document in 'Outline' view!)O. Estimation of Parameters.1. Point and Interval Estimation. Properties of Estimators. Let be an estimator for .a. Unbiassedness ˆE .b. Consistency (As sample size gets larger, estimate gets better).c. Efficiency (ˆ has a small variance). Define BLUE.d. Maximum Likelihood (ˆ is the value of that is most likely to have produced the observed data) .2. A Confidence Interval for When is Known.xzx2 You can only use this when you know the population variance. Don’t forget that there are two formulas for the standarderror depending on sample size! x is the standard error.An interval of this type is used in two situations: (i) where the population variance, 2, is in fact, known and the sample size is relatively large; or (ii) where the variance is not known and the sample variance, 2s, is used to replace 2, but the degrees of freedom 1nare so large that the appropriate value of 1nt is not very different from z. The first of these situations is not very realistic, but serves as a good introduction to confidence intervals. The formula for this type of confidence interval for the mean is, xzx2, where nx. Note: If ,05. Nn use 1NnNnx(nis sample size and Nis population size) Example 1: Assume that a population is Normally distributed with anunknown mean and a population standard deviation of 36. 36?,~ Nx From a random sample of size 9n, we get a samplemean of 62. (Because the population variance is known we can ignore any sample variance we might compute from the data.). Find a 95% confidence interval for the mean.Step 1: State the confidence level and significance level.The given confidence level of 95% represents the probability that the interval actually contains the mean and is stated as .95.1 The significance level of 5% represents the probability of beingwrong and is .05. Step 2: Find the appropriate value of .z Use the last line of Table 18 (or Table 17) in the Syllabus Supplement to find 960.1025.2zz (the bottom number in the .025 column). Note that higher confidence levels give larger values of z, and thus larger confidence intervals.Step 3: Find the standard error. .12936nx Note that larger values of n make the standard error and the confidence interval smaller.Step 4: Put it together. 12960.1622xzx52.2362 . The last part of
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