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CHAPTER 5: Statistics Topics Overview p. 174 Mean & Standard Deviation Calculate standard deviation (Sx) by hand (FORMULA) Questions: p. 176-177 #1-4 p.181-182 Normal Distribution Questions: p.182-184 #12-17 p.187-188 Sampling Methods Questions: p.189-190 #1-5 p.196-197 Central Limit Theorem Questions: p.198-199 #4-9 Confidence Intervals p.202-204 Confidence Intervals when population mean and standard deviation given Questions: p.205 #20 p.209-210 Confidence Intervals when population mean and standard deviation NOT given Questions: p.211 #28-30 Statistics Unit Detailed Summary For more complete summary, see unit summary p.234-239 Population – the set representing all measurements of interest to the investigator Sample – a subset of measurements selected from a population of interest Topics 1. Calculating Standard Deviations (Formulae p.174 & 176, Exercises1-4 pp.176-178) 2. Simulations (p.179-180)** (**We may not cover) 3. Normal Distribution (p.181-182, Exercises #12-17 p.182-184) 4. Sampling Methods (p.187-188, Exercises #1-5 p.189-190, p.244#4) Symbols x Sx Population Mean Population Standard Deviation Sample Mean Sample Standard Deviation x Mean of Sample Means x Standard Deviation of Sample Means 5. Sampling Distribution of Sample Mean (Central Limit Theorem) Values of the sample mean are unpredictable and vary from sample to sample. To make predictions about a population, we must repeatedly draw random samples of a specific size from a population. If the mean of each sample has been taken, the mean of these sample means would be approximately equal to the population mean. x = ( is the symbol for ‘mean of sample means’) x Also, the following equation represents the standard deviation of all the sample means (x) x= n ( is the standard deviation of all the sample means found, is the population standard x deviation & n in the size of the samples drawn) See Example 1 p.197, Try Exercises #3-8 p.198-199 6. Point Estimates and Confidence Intervals A simple sample mean is called a point estimate because this single number can be used as a possible estimate of the population mean. Because this is often inaccurate, we instead find an interval in which we are confident that that the population mean would be contained. This is a confidence interval. Confidence Intervals Earlier in the unit we approximated that 95% of the data falls within 2 standard deviations of the mean. In fact, 95% of the data actually falls within 1.96 standard deviations of the mean. Depending on the “percent confidence” you are looking for, you will use different z-values (i.e.1.96) to represent the number of standard deviations from the mean. _ Confidence Interval Formulae x _ z _ or x z x n Z-value chart Confidence Interval 90% 95% 99% z-value 1.645 1.96 2.56 Example: Joe collects a sample of size 35 from a known population with a population mean of 250 and a population standard deviation of 20. She finds that the sample mean is 246. Determine the 95% confidence interval for this sample. _ x z 246 n 1.96 95% Confidence Interval 20 246 35 6.6 239.4 to 252.6 Description of Answer: The mean will be between 239.4 and 252.6 95% of the time. (19 times out of 20) (see pp.201-205, Exercises #20,21 pp.205-206) Confidence Intervals When is Unknown _ We used x z to calculate confidence intervals. n It is most likely, however, that you WON’T know the population standard deviation () when calculating confidence intervals for real-life sample data. Instead, we will simply use the sample standard deviation (Sx) in place of , so long as the sample size is large enough (n30). The following formula will be used as our interval estimator: _ Confidence Interval Formula for Samples Note: The z Sx n x z Sx n part of the formula is called the margin of error.