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This 3 page Class Notes was uploaded by Kayla Ann Berube on Thursday January 21, 2016. The Class Notes belongs to Stat 110 at a university taught by Wilma Sims in Spring 2016. Since its upload, it has received 8 views.
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Date Created: 01/21/16
What Do Samples Tell Us? • Parameter o A number that describes the population. It is a fixed number, but we usually don’t know its value. • Statistic o A number that describes a sample. Its value it computed from sample information, but it can change from sample to sample. Proportions • We want to estimate the proportion of individuals in a population with a certain characteristic (“success”). • The population proportion, p, is an unknown parameter. • We wish to estimate p based on a sample. • Example: o Take a SRS of size n from a population with p successes. o p hat is the sample proportion where… count of successes in the sample § p hat = n Error in Estimation • We wish to estimate a parameter from a statistic. • Bias o Consistent, repeated deviation of the sample statistic from the population parameter in the same direction when we take many samples. • Variability o Describes how spread out the values of the sample statistic are when we take many samples. • A good sampling method has both small bias and small variability. Margin of Error • A statistic, calculated from a random sample, will usually not estimate the parameter exactly. • Surveys often report a percentage and a “margin of error”. • “Margin of error or minus 3 percentage points” means that 95% of all samples would give a result within plus or minus 3 percentage points of the true parameter. • A margin of error tells us how close our estimate comes to the truth. • Quick method for margin of error (MOE): o Use the sample proportion, p hat, from a SRS of size n to estimate an unknown population proportion p. o The margin of error for 95% confidence is approximately: § 1 divided by the square root of n Confidence Statements • We say we are 95% confident that the true value of the parameter lies within the margin or error. • Confidence Statement o Indicates how precise a statistic is by giving a degree of sureness that the actual parameter is within the range given by the statistic and its margin of error. o p hat + MOE • Hints for interpreting confidence statements: o The conclusion of a confidence statement always applies to the population, not to the sample. o Our conclusion about the population is never completely certain. o A sample survey can choose to use a confidence level other than 95%. o To decrease the width of a 95% confidence interval (i.e. have a samller margin of error), increase the sample size. Sample Surveys in the Real World • Sampling and statistics seem simple, but some problems can arise. • Sampling Errors: o Errors caused by the act of taking a sample. They cause sample results to be different from the results of a census. • Errors in Sampling: o Undercoverage § Occurs when some groups in the population are left out of the process of choosing the sample. § Sampling Frame • A list of all available members of the population from which the sample will be selected. o Random Sampling Error § Results from chance selection in the SRS. § The error is due to chance (always present). § A large sample helps control this. § The margin of error includes only random sampling error. o Nonsampling Errors § Errors not related to the act of selecting a sample from the population. They can be present even in a census. § Nonresponse (missing data) • Refusal to answer survey, subject is not available for survey. § Response Errors • Subject may lie or remember incorrectly, subject may not understand question. § Processing Error • Math errors, coding data incorrectly. § Effects of Data Collection Procedure • Wording of the question, timing/events, how survey is administered (mail, telephone, personal interview). How to Live with Errors in Sampling • Substitute other households for non-responders. • Weight the responses o Can also account for the differences by race, age, household size, etc. o Helps correct bias, but increases variability. Stratified Random Sampling Design • Step 1: Divide the sampling frame into groups of individuals called strata. o The strata are chosen using some characteristic of the individuals already known and of special interest. • Step 2: Take a separate simple random sample in each stratum and combine these to make up the stratified random sample. Probability Sample • A sample chosen in such a way that we know what samples are possible and what chance, or probability, each possible sample has to be chosen (not all need to be equally probable). • Simple random and stratified random are probability samples.
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