Estimation from Samples Find a likely range of values for a population parameter (e.g. average, %) Find a likely range of values for a population parameter (e.g. average, %) Parameter = characteristic of a population Parameter = characteristic of a population Statistic = characteristic of a sample Statistic = characteristic of a sample Statistical inference = drawing conclusions about a population based on sample data Statistical inference = drawing conclusions about a population based on sample data Usually connected with a probability of error. Usually connected with a probability of error.

Sampling Distribution Distribution of results of all possible samples of size N taken from same population Distribution of results of all possible samples of size N taken from same population Theoretical, not actually done in practice Theoretical, not actually done in practice Properties of sampling distributions are known to statisticians Properties of sampling distributions are known to statisticians Used as basis for inferring from samples to populations Used as basis for inferring from samples to populations

Example: estimating proportion of homes with internet access Suppose population proportion =.62 Suppose population proportion =.62 Take 1 sample of size 200 homes. 150 have internet access. Sample p =.60 Take 1 sample of size 200 homes. 150 have internet access. Sample p =.60 Can we conclude that the population proportion is.60? Can we conclude that the population proportion is.60? A different sample might produce a different answer A different sample might produce a different answer

What we know from sampling distribution: We DON’T know the true population proportion. We DON’T know the true population proportion. We DO know how many sample proportions fall within a given distance of the true proportion. We DO know how many sample proportions fall within a given distance of the true proportion. Sampling error = estimated difference between sample value and actual population value Sampling error = estimated difference between sample value and actual population value (example: 95% of sample proportions fall within + 3% of true proportion)

Most sample proportions would be close to population value Most sample proportions would be close to population value A few would be much higher or lower A few would be much higher or lower Average of sample proportions would be the true population proportion Average of sample proportions would be the true population proportion Distribution would be a bell-shaped curve Distribution would be a bell-shaped curve What if we took all possible samples? % of samples All possible sample proportions Each section = 1 standard error 95% of p’s 68% of p’s

How we make an estimate Find sample proportion Find sample proportion Add (margin of error) on either side = 2(S.E.) Add (margin of error) on either side = 2(S.E.) True proportion falls within this interval for 95 % of all possible samples True proportion falls within this interval for 95 % of all possible samples % of samples All possible sample proportions p p p % of samples All possible sample proportions p p p

Examples of estimates If 95% of sample proportions (p) fall within + 6% of true proportion, then 95% of all intervals p +.06 will contain true population proportion. If 95% of sample proportions (p) fall within + 6% of true proportion, then 95% of all intervals p +.06 will contain true population proportion. If p =.6, we estimate the true proportion is =.54 to.66 (correct) If p =.6, we estimate the true proportion is =.54 to.66 (correct) If p =.62, we estimate the true proportion is =.56 to.68 (correct) If p =.62, we estimate the true proportion is =.56 to.68 (correct) If p =.57, we estimate the true proportion is =.51 to.63 (correct) If p =.57, we estimate the true proportion is =.51 to.63 (correct) If p = If p =.7, we estimate the true proportion is =.64 to.76 (incorrect) If p = If p =.7, we estimate the true proportion is =.64 to.76 (incorrect) 95% of the time this procedure yields a correct estimate.