Sampling and Sampling Distribution

Sampling and Sampling Distribution Sampling -a major portion of statistics is concerned with statistical inference, where we examine the problem of estimating population parameters or testing hypotheses about such parameters -a population consists of all items of interest in the statistical problem -sample is a subset of the population -sample statistic is used to make inferences about the unknown population parameter -bias refers to the tendency of a sample statistic to systematically overestimate or underestimate a population parameter -often caused by samples that are not representative of the population -selection bias refers to a systematic underrepresentation of certain groups from consideration for the sample -nonresponse bias refers to a systematic difference in preferences between respondents and nonrespondents to a survey or a poll -social-desirability bias refers to a systematic difference between a group’s “socially acceptable” responses to a survey or poll and this group’s ultimate choice -a simple random sample is a sample of n observations that has the same probability of being selected from the population as any other sample of n observations, most statistical methods presume simple random sampling -in stratified random sampling, the population is first divided up into mutually exclusive and collectively exhaustive groups, called strata, including random selected observations from each strata -preferred when the objective is to increase precision -cluster sampling, another method for forming a representative sample, is formed by dividing the population into mutually exclusive and collectively exhaustive groups called clusters, observations from randomly selected clusters -works best when most of the variation in the population is within the groups and not between the groups -prefered when the objective is to reduce costs The Sampling Distribution of the Sample Mean -a parameter is a constant -a statistic is a random variable whose value depends on the chosen random sample -when a statistic is used to estimate a parameter, it is referred to as an estimator -a particular value of the estimator is called an estimate -an estimator is unbiased if its expected value equals the population parameter -the sampling distribution of the mean is the probability distribution derived from all the means that come from all possible samples of a given size -consider a sample mean derived from n observations -another sample mean can be derived from a different sample of n observations -repeat the process a large number times -the frequency distribution of the sample means is the sampling distribution - is a random variable -the X represents a certain characteristic of a population -population mean E(X) = , variance Var(X) = -let the sample mean be based on a random sample of n observations -let X = value when roll a die -what does the distribution of X look like? -it is discrete uniform distribution - = 3.5 - = 1.7 -instead of rolling a die once, what if we roll a die 50 times and record the mean, and then we repeat the process 100 times? we will have 100 sample means each from a sample size n = 50 -what will the distribution of look like? -occasionally we may get a run of small numbers or a run of large numbers, but we expect most of the sample means to be close to 3.5 -in fact while the range of x values is 1 to 6, the range of will be smaller, it is very unlikely we would get an (or even close) or get an = 6 -the expected value of is the same as the expected value of X -E() = E(X) = -the average of the sample means is the average of the population mean -unbiased: expected value of an estimator equals the population Parameter -the variance of is Var() = -Var() is less than Var(X) = -each sample will contain both high and low values that cancel on another -the standard deviation of is the standard error: sd() = -for making statistical inferences, it is essential that that sampling distribution of is normally distributed -what if the underlying population is not normally distributed? -the central limit theorem (CLT) states that the sum or the average of a large number of independent observationsfrom the same underlying distribution has an approximate normal distribution -the approximation steadily improves as the number of observations increases -practitioners often use the normal distribution approximation when n≥ 30 -note E() = E(X) = and sd() = -central limit theorem summary -if the original population is normally distributed, the sampling distribution of the mean will also be normally distributed -if the original population is not normally distributed but symmetric, for samples of size n greater than 15, the sampling distribution of the mean can be approximated by a normal distribution, the approximation gets better as the sample size increases -if the original population is not normally distributed and not symmetric, for samples of size n greater than 30, the sampling distribution of the mean can be approximated by a normal distribution, the approximation gets better as the sample size increases -means of the sample means -standard deviation of the sample means -use the original probability distribution with the given population mean and standard deviation when asked about probability of an individual value -use the CLT method when asked about the probability of the mean of more than 1 value -example: tests scores are normally distributed with a mean of 80.3 and a standard deviation of 5.2 -if an individual test score is randomly selected, what is the probability the score is greater than 92? P(X > 92) > pnorm(92, 80.3, 5.2, FALSE) > 0.01222447 -if 15 scores are randomly selected, what is the probability that their mean teast score is greater than 92? P( > 92) MUST USE CLT! > pnorm(92, 80.3, 5.2/sqrt(15), FALSE) > 1.463933e-18 -if an individual test score is randomly selected, what is the probability that the score is between 75 and 85? P(75 < X < 85) > pnorm(85, 80.3, 5.2) - pnorm(75, 80.3, 5.2) > 0.6629148 -if 15 scores are randomly selected, what is the probability that their mean test score is between 75 and 85? P( 75 < < 85) > pnorm(85, 80.3, 5.2/ sqrt(15)) - pnorm(75, 80.3, 5.2/ sqrt(15)) > 0.9997284 The Sampling Distribution of the Sample Proportion -in many business applications, we are concerned with the population proportion p -recall that the binomial distribution describes the number of successes X in n trails of a Bernoulli process where p is the probability of success -use the sample proportion -E() = p; so the sample proportion is unbiased -se() = -by the CLT, the sampling distribution of is approximately normal when np ≥ 5 and n(1-p) ≥ 5 -transform into its corresponding z value -example: a study found that 55% of british firms experienced a cyberattack in the past year -what are the expected value and the standard error of the sample proportion derived from a random sample of 100 firms? p = 0.55 and n = 100 E() = 0.55 se() = -what are the expected value and the standard error of the sample proportion derived from a random sample of 200 firms? p = 0.55 and n = 200 E() = 0.55 se() = -comment on the value of the standard error as the sample size gets larger -as the sample size gets larger, the standard error gets smaller -if we randomly select 100 firms, what is the probability more than 60% have experienced a cyber attack? p = 0.55, find p( > 0.60) > pnorm(.60, .55, sqrt(.55*(1-.55)/100), FALSE) > 0.1574393 -if we randomly select 100 firms, what is the probability that less than 45% have experienced a cyber attack? p = 0.55, find p( < 0.45) > pnorm(.45, .55, sqrt(.55*(1-.55)/100)) > 0.02221159 sampling distribution of when the population proportion is p = 0.10 sampling distribution of when the population proportion is p = 0.30 the weight of people in a small town in Missouri is known to be normally distributed with a mean of 176 pounds and a standard deviation of 29 pounds. on a raft that takes people across the river, a sign states, “maximum capacity 3,383 pounds or 17 persons.” what is the probability that a random sample of 17 persons will exceed the weight limit of 3,383 pounds? -think about in terms of CLT -the probability that the total weight of the 17 selected is greater than 3,383 pounds is the same as the probability that their average weight is greater than 3,383/17 = 199 pounds per person P( > 199) = P= P(Z > 3.27) = 1-0.9995 = 0.0005 > pnorm(199, 176, 29/ sqrt(17), lower.tail = FALSE) > 0.0005376437 -40% of regular customers from a restaurant prefer to not have to make a reservation. if 35 customers are randomly selected, what is the probability more than 50% prefer to not have to make a reservation? p = 0.40, find P( > 0.50) P > pnorm(.50, .40, sqrt(.40*1-.40/35), FALSE) > 0.1136