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  Sample standard deviation is the most common estimator for  (s 2 is unbiased for  2) 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  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 (  is also a parameter). It is estimated by replacing  with its estimate from the sample data: 95% Confidence Interval for  :

Confidence Interval for the Mean Most reported confidence intervals are 95% By increasing confidence coefficient, width of interval must increase Rule for (1-  )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  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:  (a value between 0 and 1, not ) Sample - n items sampled, X is the number that possess the characteristic (fall in the category) Sample Proportion: –Mean of sampling distribution:  –Standard error (actual and estimated):

Confidence Interval for a Proportion Criteria for large samples – –Otherwise, X > 10, n-X > 10 Large Sample (1-  )100% CI for  :

Choosing the Sample Size Bound on error (aka Margin of error) - For a given confidence level (1-  ), we can be this confident that the difference between the sample estimate and the population parameter is less than z  /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-  ) Proportions (can be safe and set  =0.5): Means (need an estimate of  ):

Small-sample Inference for  t Distribution: –Population distribution for a variable is normal –Mean , Standard Deviation  –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 t A where P(t > t A ) = A for degrees of freedom 1-29 and various A

Probability DegreesofFreedomDegreesofFreedom CriticalValuesCriticalValues Critical Values

Small-Sample 95% CI for  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 - 50 th -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