Chapter 9 Sampling distributions Parameter – number that describes the population A parameter is a fixed number, but in reality we do not know its value because we can not examine the entire population. Statistic – number that describes a sample Use statistic to estimate an unknown parameter. μ = mean of population x = mean of the sample Sampling variability – the differences in each sample mean P(p-hat) - the proportion of the sample 160 out of 515 people believe in ghosts P(hat) = = .31 .31 is a statistic Proportion of US adults – parameter Sampling variability Take a large number of samples from the same population Calculate x or p(hat) for each sample Make a histogram of x(bar) or p(hat) Examine the distribution displayed in the histogram for shape, center, and spread as well as outliers and other deviations Use simulations for multiple samples – much cheaper than using actual samples Sampling distribution of a statistic is the distribution of values taken by the statistics in all possible samples of the same size from the same population Describing sample distributions Ex 9.5 page 494-495, 496 Bias of a statistic Unbiased statistic- a statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. Ex 9.6 page 498 Variability of a statistic is described by the spread of its sampling distribution. The spread is determined by the sampling design and the size of the sample. Larger samples give smaller spreads!! Homework read pages 488 – 503 do problems 1-4, 12- 15 9.2 Sampling proportions P(hat) = = Mean of sampling distributions μp(hat) = p standard deviation of sampling distribution σp(hat) = Must always check these two items!!!!!!!!!!!! Rule of thumb 1- population must be at least 10x greater than sample size (used to see if you can use standard deviation formula) Rule of thumb 2 – we will use normal approximation as long as np 10 n(1-p) 10 example 9.7 Given μp(hat) = .35 n = 1500 What percent fall with 2% of the true value? ______________________________ .33 .35 .37 (check for rule 1) σp(hat) = σp(hat) = = .0123 check to see if we can use normal distribution np 10 n(1-p) 10 1500(.35) 10 1500(.65) 10 Yes yes Check is population > 10 times 1500? Yes Need to find z-scores .33-.35 .37 - .35 .0123 .0123 = -1.63 = 1.63 .9484 .0516 ____________________________ .33 .37 .9484 - .0516 = .8968 89.68% fall within 2% of the true value P(.33 p(hat) .37) Homework read pages 504 – 512 problems 19 – 23 9.3 Sampling μ = mean of population x = mean of sample μx = μ σx = (when taking a sample and finding the standard deviation) N(μ, ) normal distribution of a sample mean Z = (when sample size is 1) Z = (when sample size is greater than 1) Ex. X = height of randomly selected woman μ = 64.5 inches σ = 2.5 inches take a sample of size 10 find σx = = = .79 p(x > 66.5) z = = 2.53 .9943 .0057 __________________ 0.57 % chance that an SRS of 10 women would have an average height of 66.5 inches or taller .7881 .2119 Individual person Z = = .8 ______________________ 21.19 % chance of a woman being taller than 66.5 inches. Do not forget sample of size n z = Homework problems 32, 33, 39 Central limit theorem Draw an SRS of size n from any population what so ever with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x is close to normal N(μ, ). Page 522 shows examples of graphs Homework problems 45, 47 – 53 Take home quiz .