Toward statistical inference Sampling variability Toward statistical inference

Parameter vs. statistic Number that describes a population Ex: mean, variance Learn about it by taking sample - statistical inference Number that describes a sample Value is known after examining the sample Value changes from sample to sample

Sampling variability If you take a simple random sample from a population, every sample will likely be different. Thus, the value of a statistic varies with repeated random sampling. If we take many such samples, we notice an emerging pattern - the sampling distribution

Sampling distribution Portrays how the statistic changes from sample to sample We can get an idea of it by Taking many random samples of the same size, n, from the same population Calculating the sample proportion for each sample Examining a histogram of all the sample proportions obtained.

Estimating population proportion What proportion of the population meets a certain criterion? View this as people/objects marked with “1” (if meet criterion) and “0” (if don’t meet criterion) Take sample, add up the “1”s, divide by n - yields sample proportion

Population proportion (real life examples) What percentage of California voters voted for Arnold? What percentage of items coming off the assembly line have a certain defect? What percentage of seeds (in an agricultural field trial) sprouted?

Simulation To understand shape of sampling distribution, we can use simulation Use computer to take random draws from population, graph results For population of “0”s and “1”s, we can use the computer to generate from a binomial distribution, where n is sample size and p is population percentage meeting our criterion

Mean of a statistic The mean is the center of the sampling distribution for the statistic. It is the average value of the statistic over many samples. If the mean of the sampling distribution is equal to the parameter of interest, the statistic is unbiased.

Variability of a statistic What is the spread of the sampling distribution? Spread is smaller for larger sample sizes - more data helps you understand the parameter better Variability of a statistic is unaffected by the size of the population as long as the population is much bigger than the sample size, n.

Goldenrod and the gall fly

Cautions The sampling distribution only shows how the statistic varies due to chance. Sampling distribution doesn’t show possible bias due to undercoverage, nonresponse, lack of realism, etc.