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Estimation • Goal: Use sample data to make predictions regarding unknown population parameters • Point Estimate - Single value that is best guess of true parameter based on sample • Interval Estimate - Range of values that we can be confident contains the true parameter Point Estimate • Point Estimator - Statistic computed from a sample that predicts the value of the unknown parameter • Unbiased Estimator - A statistic that has a sampling distribution with mean equal to the true parameter • Efficient Estimator - A statistic that has a sampling distribution with smaller standard error than other competing statistics Point Estimators • Sample mean is the most common unbiased estimator for the population mean m Y m Y ^ i n • Sample standard deviation is the most common estimator for s (s2 is unbiased for s2) ^ s s 2 ( Y Y ) i n 1 • Sample proportion of individuals with a (nominal) characteristic is estimator for population proportion Confidence Interval for the Mean • Confidence Interval - Range of values computed from sample information that we can be confident contains the true parameter • Confidence Coefficient - The probability that an interval computed from a sample contains the true unknown parameter (.90,.95,.99 are typical values) • Central Limit Theorem - Sampling distributions of sample mean is approximately normal in large samples Confidence Interval for the Mean • In large samples, the sample mean is approximately normal with mean m and standard error sY s n • Thus, we have the following probability statement: P(m 1.96s Y Y m 1.96s Y ) .95 • That is, we can be very confident that the sample mean lies within 1.96 standard errors of the (unknown) population mean Confidence Interval for the Mean • Problem: The standard error is unknown (s is also a parameter). It is estimated by replacing s with its estimate from the sample data: s sY n ^ 95% Confidence Interval for m : ^ Y 1.96 s Y s Y 1.96 n Confidence Interval for the Mean • Most reported confidence intervals are 95% • By increasing confidence coefficient, width of interval must increase • Rule for (1-a)100% confidence interval: s Y za / 2 n (1-a)100% 90% 95% 99% a .10 .05 .01 a/2 .050 .025 .005 za/2 1.645 1.96 2.58 Properties of the CI for a Mean • Confidence level refers to the fraction of time that CI’s would contain the true parameter if many random samples were taken from the same population • The width of a CI increases as the confidence level increases • The width of a CI decreases as the sample size increases • CI provides us a credible set of possible values of m with a small risk of error Confidence Interval for a Proportion • Population Proportion - Fraction of a population that has a particular characteristic (falling in a category) • Sample Proportion - Fraction of a sample that has a particular characteristic (falling in a category) • Sampling distribution of sample proportion (large samples) is approximately normal Confidence Interval for a Proportion • Parameter: p (a value between 0 and 1, not 3.14...) • Sample - n items sampled, X is the number that possess the characteristic (fall in the category) X p • Sample Proportion: n ^ – Mean of sampling distribution: p – Standard error (actual and estimated): ^ p 1 p n ^ s ^ p p (1 p ) n ^ s ^ p Confidence Interval for a Proportion • Criteria for large samples – 0.30 < p < 0.70 n > 30 – Otherwise, X > 10, n-X > 10 • Large Sample (1-a)100% CI for p : ^ p 1 p n ^ ^ p za / 2 Choosing the Sample Size • Bound on error (aka Margin of error) - For a given confidence level (1-a), we can be this confident that the difference between the sample estimate and the population parameter is less than za/2 standard errors in absolute value • Researchers choose sample sizes such that the bound on error is small enough to provide worthwhile inferences Choosing the Sample Size • Step 1 - Determine Parameter of interest (Mean or Proportion) • Step 2 - Select an upper bound for the margin of error (B) and a confidence level (1-a) Proportions (can be safe and set p=0.5): Means (need an estimate of s): 2 za / 2 p (1 p ) n B2 2 za / 2 s 2 n B2 Confidence Interval for Median • Population Median - 50th-percentile (Half the population falls above and below median). Not equal to mean if underlying distribution is not symmetric • Procedure – – – – Sample n items Order them from smallest to largest n 1 n Compute the following interval: 2 Choose the data values with the ranks corresponding to the lower and upper bounds