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Topic 12 Sampling Distributions Sample Proportions • is determined by: = successes / size of sample = X/n If you take as SRS with size n with population proportion p, then the mean of the sampling distribution is exactly p. • • o This means that is an unbiased estimator of p. The standard deviation of the sampling distribution is o Only use this if the population is ten times the sample size. To determine if the sampling distribution of o np > 10 and o n(1-p) > 10 is normal: Sample Means If is the mean of an SRS size n from population with mean and standard deviation , then: o The mean is o The standard deviation is • • o Sample mean is unbiased estimator of population mean Larger samples = less spread o Standard deviation decreases at a rate of the , so you must take a sample 4 times as large to cut the standard deviation in half Only use for the standard deviation of if the population is at least 10 times the size of the sample Central Limit Theorem • An SRS that is large enough (~ >30) can be • • considered normally distributed If the population distribution is very skewed, it takes a very large SRS to use the central limit theorem The CLT allows us to use normal probability calculations even when the population distribution is not considered to be normal Topic 13 Confidence Intervals Confidence Intervals Confidence intervals estimate the true value of the parameter where the parameter is the true mean , true proportion p, or true slope . estimate (critical value)(sta ndard error of the estimate) Confidence Intervals 1-sample t-interval for µ 2-sample t-interval for µ1 - µ2 Matched-pairs t-interval 1-proportion z-interval for p 2-proportion z-interval for p1 - p2 t-interval for slope Interpret the confidence level: C% of all intervals produced using this method will capture the true mean (difference in means), or proportion (difference in proportions), or slope. (Describe the parameter in context!) Interpret the confidence Interval: I am C% confident that the true parameter (insert context) is between ___ and ___ (insert units), based on this sample. What does it mean to be 95% confident? • 95% chance that is contained in the confidence interval • The probability that the interval contains is 95% • The method used to construct the interval will produce intervals that contain 95% of the time. Margin of error • Shows how accurate we believe our estimate is • The smaller the margin of error, the more precise our estimate of the true parameter • Formula: critical me value standard deviation of the statistic How can you make the margin of error smaller? • z* smaller (lower confidence level) • s smaller (less variation in the population) • n larger Really cannot change! (to cut the margin of error in half, n must be 4 times as big) Find a sample size: • If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: s me z * n Always round up to the nearest person! The heights of MRHS male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval? 2.5 0.75 1.96 n n = 42.68 or 43 students The 4-Step Process (from the Inference Toolbox) Step 1 (Population and parameter) Define the population and parameter you are investigating Step 2 (Conditions) Do we have biased data? Random? • If SRS, we’re good. Otherwise PWC (proceed with caution) Do we have independent sampling? • If pop>10n, we’re good. Otherwise PWC. Do we have a normal distribution? • If pop is normal (np>10, nq>10 or n>30 (CLT), we’re good. Otherwise, graph it (histogram!). The 4-Step Process (from the Inference Toolbox) Step 3 (Calculations) • Find z* or t * based on your confidence level (and df). If you are not given a confidence level, use 95% • Calculate CI. Step 4 (Interpretation) • “We are ______% confident that the true mean (or proportion or slope) is captured in the interval (lower, upper)” and don’t forget CONTEXT!!!!! How does t compare to normal? • Shorter & more spread out • More area under the tails • As n increases, t-distributions become more like a standard normal distribution How to find t* Can also use invT on the calculator! • Use Table B for t distributions Need up upper t* value with above & – so • Look confidence level5% at is bottom df on 95% is below the sides • df = n – 1 invT(p,df) Find these t* 90% confidence when n = 5 95% confidence when n = 15 t* =2.132 t* =2.145