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2 Pages

### Sp06SampleProbabilitiesHandout

Course: RM 1, Fall 2009
School: UNC Asheville
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Word Count: 838

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Handout: Information Using the Normal Distribution to Estimate Percentages and Probabilities (When population parameters are known) The purpose of this sheet is to provide a summary of the steps involved in calculating percentages and probabilities using the assumption of a normal distribution, both when population parameters are known (i.e. when one knows the population mean and standard deviation), and when...

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Handout: Information Using the Normal Distribution to Estimate Percentages and Probabilities (When population parameters are known) The purpose of this sheet is to provide a summary of the steps involved in calculating percentages and probabilities using the assumption of a normal distribution, both when population parameters are known (i.e. when one knows the population mean and standard deviation), and when population parameters are unknown (and can only be estimated). Following the instructions and information, exercise problems will be presented. I. When population parameters (population mean and population standard deviation) are known. A. When a sample is drawn from a population of people where the population mean and standard deviation are known (like standardized tests, etc.), various probabilities and percentages can be calculated using the table of the normal distribution. The questions that can be asked and answered are like the following: 1. What percentage of scores in the sample are below a given raw score or z-score? (This is known as a percentile estimation.) 2. If a person or a sample of people is drawn at random from the population, what are the chances that the person (or sample mean) will be X far (or farther) from the population mean? 3. What percent of the distribution lies between any two raw scores (or z-scores) centered on the mean, or what percent lies outside (in the tails of the distribution) beyond a central segment of the population that is centered on the mean? B. The steps to answering these questions (when population parameters are known) involve the same logic as our previous exercise involving the probabilities and percentages when one score is involved, except that now we are dealing with a sample of scores with sample size = N. When dealing with samples, we have to calibrate our calculations according to sample size, because the larger the sample the smaller the expected sampling error. The statistic known as the Standard Error of the Mean is what does this. The Standard Error of the mean is calculated as (Pop. SD) divided by the (square root of the sample size). Note that if your sample size is 1 (N = 1) then the square root of 1 is 1 and the Standard Error of the Mean is the same value as the Population Standard Deviation. So you can think of the Standard Error of the Mean as a different version of the Standard Deviation that is simply calibrated to your sample size. Once you calculate the Standard Error of the Mean, then you can treat it like it is a standard when deviation you calculate z-scores, percentages under the normal curve, etc., using the table of the normal distribution. C. Here are the steps along with an example: Given Information + Example Question: Let's say we have a population of IQ scores with a population mean of 100 and a population standard deviation of 15 (some of the more commonly-used IQ tests are standardized to have these population values). Question: What is the probability that a random sample of 25 people will have a sample mean that is 8 or more IQ points away from the population mean? Step 1: Calculate the Standard Error of the Mean SEmean = pop SD/(sqrt(N)) = 15/5 = 3.00 Step 2: Use the SEmean to calculate the z-scores for the upper bound and the lower bound Zupper bound = (Upper Interval Raw Score Population Mean) / SEmean = (108 100) / 3.00 = 2.67 Zlower bound = (Lower Interval Raw Score Population Mean) / SEmean = (92 100) / 3.00 = - 2.67 Step 3: Use table of the normal distribution to get the appropriate probability value (a) If the question is asking for the probability of sample means in the tails (beyond a central range): (1) Look in column `a' for the z-score of your upper bound (2) Then read the corresponding probability/proportion in column `c' (3) Turn the proportion into a percent (4) Then multiply times 2 because the same percent is in the other tail. Example: The present example is asking for the probability of a sample mean beyond a central range, so the above steps 1-4 apply. Looking in column `a' for the...

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