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+ Statistics: Chapter 6 Z scores review, Normal Curve Introduction + Do Now: Complete the Beijing Olympics worksheet given to you when you entered class. This will be an extended do now and I will collect it. You may work in table groups + Normal Models Normal Models are appropriate for distributions whose shape is uni-modal and symmetric. “Bell shaped curve” + Symbols N(mean, standard deviation) The normal model with mean 0 and standard deviation of 1 is called the standard normal model (or standard normal distribution) + Symmetric Uni-Model Data So we said that symmetric, unimodal data can be standardized into a normal model…. 1. So don’t claim a normal model with skewed data. 2. Is anything actually ever completely normal? + Normal Models The 68 – 95- 99.7 Rule (Empirical Rule) + + + + Sketch Normal Models using the 6895-99.7 Rule Birth weights of babies N(7.6 lb, 1.3 lb) + Sketch Normal Models using the 6895-99.7 Rule ACT Scores at a certain college, N(21.2, 4.4) + Do Now Create a list in your calculator with the following numbers: 4 3 10 12 8 9 3 + Calculating Standard Deviation on a Calculator Put all your data into a list Under the STAT CALC menu, select 1-VAR STATS and hit ENTER Specify the location of your data, created a command like 1VAR STATS L1. Hit ENTER again + Normal Models The 68 – 95- 99.7 Rule (Empirical Rule) + + Example: This is a practice problem in your packet*** A forester measured 27 of the trees in a large wood that is up for sale. He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches. Suppose that these trees provide an accurate description of the whole forest and the normal model applies + N(36, 4) What percent of data are between 32 and 40? What percent above the mean are between 36 and 40? What percent of data are between 28 and 44? What range contains 99.7% of data? What range contains 47.5% below the mean? Where would the top 16% of data be? What percent of data is outside 24 and 48? + Helmet Sizes The army reports that the distribution of head circumference among male soldiers is approximately normal with a mean of 22.8 inches and a standard deviation of 1.1 inches. What percent of soldiers have a head circumference greater than 23.9in? A head circumference of 23.9 inches would be what percentile? What percent of soldiers have a head circumference between 20.6 and 23.9inches? What interval below the mean would contain 13.5% of soldiers? + Driving Speed Suppose it takes you 20 minutes, on average, to drive to school with a standard deviation of 2 minutes. Suppose a normal model is appropriate for the distribution of driving times. How often will you arrive at school in less than 22 minutes? How often will it take you more than 24 minutes? Do you think the distribution of your driving times is unimodal and symmetric in general? What does this say about the accuracy of your predictions? Explain. + Practice Try some on your own! As always, call me over if you are confused! + Exit Ticket A company that manufactures rivets believes the shear strength (in pounds) is modeled by N(500, 50). Draw and label the normal model (just sketch the curve) Would it be safe to use these rivets in a situation requiring a shear strength of 750 pounds? Explain. About what percent of these rivets would you expect to fall below 900 pounds? Rivets are used in a variety of applications with varying shear strength requirements. What is the maximum shear strength for which you would feel comfortable approving this company rivets? Explain your reasoning.