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 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

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 this method on a random sample will contain the true unknown fixed 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 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

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 :

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:

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

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 :

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):

Small-sample Inference for m t Distribution: Population distribution for a variable is normal Mean m, Standard Deviation s The t statistic has a sampling distribution that is called the t distribution with (n-1) degrees of freedom: Symmetric, bell-shaped around 0 (like standard normal, z distribution) Indexed by “degrees of freedom”, as they increase the distribution approaches z Have heavier tails (more probability beyond same values) as z Table B gives tA where P(t > tA) = A for degrees of freedom 1-29 and various A

Probability Cri t ical Values Degrees of Freedom Critical Values

Small-Sample 95% CI for m Random sample from a normal population distribution: t.025,n-1 is the critical value leaving an upper tail area of .025 in the t distribution with n-1 degrees of freedom For n  30, use z.025 = 1.96 as an approximation for t.025,n-1

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