Chapter 10 Sampling and Sampling Distributions 10 1

Chapter 10 Sampling and Sampling Distributions • • • 10. 1 Random sampling 10. 4 Stratified sampling 10. 6 Sampling distribution 10. 7 The standard error of the mean 10. 8 The central limit theorem

10. 1 Random sampling • Example: At a parts depot the inventory shows 1000 parts in stock. What percentage of those are actually in stock? – Let p = true % – To estimate p, take a sample of n parts to check out in the supply room. – Then – How do we choose the parts to check?

Random sample • A random sample is an insurance policy to protect against bias. • A simple random sample gives each of the possible sample choices the same chance of being selected.

Random sampling In other words, for a finite population of N sample points: A sample of size n is random if any n sample points have a probability to be selected.

To generalize For an infinite population, or a finite population but sampling with replacement. A value is observed according to a probability distribution. A random sample of size n consists of observed values that are independent and have the same distributions.

10. 2 &10. 3 Skip

10. 4 Stratified sample • A simple random sample is not always the best option. • Consider the following example. We want to estimate the average number of vireos per area. Conifer Desiduous

Then the population mean and sample mean are Question: if we use simple random sampling, is always a good estimator of ?

• Answer: NO. • Reasoning: A simple random sample of n=10 locations might all end up Conifer. We would be better off putting n 1 samples in conifer and n 2 samples in Desiduous.

Stratified sampling • A stratified random sample breaks the population into strata and samples randomly within each stratum. • In our previous example: – Stratum 1 = Deciduous – Stratum 2 = Conifer

Notations N 1=number of units in stratum 1 N 2=number of units in stratum 2 N = N 1+N 2 = number of units in entire population n 1=number of units sampled from stratum 1 n 2=number of units sampled from stratum 2 n = n 1+n 2 = number of sampled units

Optimal allocation • How do we decide n 1 and n 2 (when n is fixed)? N 1=# of possible sampling locations in conifer N 2=# in desiduous σ1=standard deviation of bird counts in conifer σ2=standard deviation of bird counts in desiduous Optimal allocation: (Problem 10. 29) Sample more in strata, with --more units (area) --higher variability

Proportional allocation • Proportional allocation doesn’t consider σ and makes the sample number from each stratum proportional to the size of the stratum.

Example of proportional allocation If N 1=100 Then N 2=300 n=40

A comparison between simple random sampling and stratified sampling Example: Population: Weights of rocks 4, 6, 10, 12. So N=4 and m=8. Sample n=2. Possible results for simple random samples Sample 4 6 4 10 4 12 6 10 6 12 10 12 Probability 1/6 1/6 1/6 sample mean 5 7 8 8 9 11 Probability 1/6 2/6 1/6

Example Continued • Possible results for a stratified sample. Choose 1 from the smaller rocks and choose 1 from the larger rocks. Sample 4 10 4 12 6 10 6 12 Probability 1/4 1/4 sample mean 7 8 8 9 Probability 1/4 1/2 1/4

A comparison between simple random sampling and stratified sampling SRS Stratified The stratified sample is more likely to be close to the true population value, m=8 here.

10. 6 Sampling Distributions A statistic (e. g. sample mean) from a random sample or randomized experiment is a random variable and its probability distribution is a sampling distribution. The population distribution of a variable is the distribution of its values for all members of the population. The population distribution is also the probability distribution of the variable when choosing one subject at random from the population.

Sampling Variability Approximating the Sampling Distribution Results of many random samples of size n=100, from a population where it is known that 60% of the people hate to shop for clothes, population proportion (parameter) p =. 6) and sample proportion (statistic). Most samples of size n = 100 gave estimates close to. 6, but some were far off. On average, they centered on. 6, they are variable, but unbiased.

Sampling Variability (cont. ) Approximating the Sampling Distribution Results of many random samples of size n=2500, from the same population, with population proportion (parameter) p =. 6) and sample proportion (statistic). Larger samples are more precise (have less variability) AND are unbiased.

Sample size What advantage is there of taking a larger sample? Larger n? Taking a larger sample decreases the potential deviation of away from m. Let be the standard deviation of the sampling distribution of , then the larger the sample size is, the smaller is.

Unbiased estimators • P. 77 “Estimators having the desirable property that their values will on average equal the quantity they are supposed to estimate are said to be unbiased. ” • is an unbiased estimator of m. • Another example of unbiased estimator is s 2 for s 2. • Choosing non-random samples can introduce bias.

10. 7 The Standard Error of the Mean If x is the mean of a sample of size n from a population having mean m and standard deviation s. The mean and standard deviation of x are: (standard error of the mean) for finite populations So, as n increases, decreases. If the sample size is multiplied by 4, the “standard deviation” (standard error of the mean) will be divided in half for infinite populations.

Standard Deviation vs. Standard Error of the Mean • The standard error of the mean is smaller than the standard deviation by a factor of the square root of the sample size. • The standard deviation describes the variability of individuals. • The standard error of the mean describes the accuracy of means of a given size or the potential error in sample mean as a guess at m.

Examples • Population size N=4, sample size n=2, 110 112 111150 152 What is the standard error of the mean?

Examples • Infinite Population P(3)=P(5)= P(7)=P(9)= 1/4 What is the standard error of the mean?

Example • Governor’s Poll 1, 1, 1 0, 1, 0 0, … What is the standard error of the mean? Estimate guess

If a population has the N(m, s) distribution, then the sample mean x of n independent observations has the distribution. 10. 8 Central Limit Theorem For ANY population with mean m and standard deviation s the sample mean of a random sample of size n is approximately when n is LARGE.

Central Limit Theorem If the population is normal or sample size is large, sample mean follows a normal distribution and follows a standard normal distribution.

• The closer x’s distribution is to a normal distribution, the smaller n can be and have the sample mean nearly normal. • If x is normal, then the sample mean is normal for any n, even n=1.

Central Limit Theorem At Work n=1 n=2 n=10 n=25

• Usually n=30 is big enough so that the sample mean is approximately normal unless the distribution of x is very asymmetrical. • If x is not normal, there are often better procedures than using a normal approximation, but we won’t study those options.

Example • X=ball bearing diameter • X is normal distributed with m=1. 0 cm and s=0. 02 cm • =mean diameter of n=25 • Find out what is the probability that will be off by less than 0. 01 from the true population mean.

Exercise The mean of a random sample of size n=100 is going to used to estimate the mean daily milk production of a very large herd of dairy cows. Given that the standard deviation of the population to be sampled is s=3. 6 quarts, what can we assert about he probabilities that the error of this estimate will be more then 0. 72 quart?