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Estimation

Learn about point estimation, interval estimation, unbiased estimators, confidence intervals, sample proportions, and more to make informed predictions and infer unknown population parameters using statistical data. Understanding the principles and methods of estimation is crucial in data analysis and decision-making processes.

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Estimation

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  1. 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

  2. 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

  3. Point Estimators • Sample mean is the most common unbiased estimator for the population mean m • Sample standard deviation is the most common estimator for s (s2 is unbiased for s2) • Sample proportion of individuals with a (nominal) characteristic is estimator for population proportion

  4. 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

  5. Confidence Interval for the Mean • In large samples, the sample mean is approximately normal with mean m and standard error • Thus, we have the following probability statement: • That is, we can be very confident that the sample mean lies within 1.96 standard errors of the (unknown) population mean

  6. 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: 95% Confidence Interval for m :

  7. 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:

  8. 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

  9. 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

  10. 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) • Sample Proportion: • Mean of sampling distribution: p • Standard error (actual and estimated):

  11. 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 :

  12. 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

  13. 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):

  14. 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 • Compute the following interval: • Choose the data values with the ranks corresponding to the lower and upper bounds

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