Notes 12.8: Normal Distribution For many populations, the distribution of events are relatively close to the average or mean. The further you go out both above and below the mean, there are fewer number of events. A Normal Distribution, or Bell Curve, is one such distribution that mirrors many statistical situations.
Properties of a Normal Distribution: 1. The curve is bell shaped with the highest point above the mean 2. The distribution is symmetrical about the center (which is the mean, median and mode) 3. You know the probability of any range of values for the distribution if you know the mean and the standard deviation for the distribution. The sum of these values (the area under the curve) is 1. 4. Because of 1 and 2, the mean, median and mode are all the same point. *Note: the standard deviation (sigma; ) is a measure used to quantify the amount of variation or dispersion of a set of data. More to come in AP Statistics or Stats 101 in college! :)
The Empirical Rule: approximates the probability that an event falls within a certain range of values (the area under the bell curve) given that the data is normally distributed The empircal rule is: 1. 68% of the data falls within 1 standard deviation of the mean 2. 95% of the data falls within 2 standard deviations of the mean 3. 99.7% of the data falls within 3 standard deviations of the mean. [Note: these values are only approximate. The exact percentages and how to determine them is beyond the scope of this course, but is covered in a senior level statistics class]. μ (pronouced "mu") = mean σ (pronounced "sigma") = standard deviation
1. A study was conducted to study the weights of high school wrestlers. It was found that the mean weight was 160 pounds with a standard deviation of 20 pounds. What percentage of wrestlers would fall between 120 180 pounds? 0.15% 0.15% Suggestions to Solve: 1. Make your own bell curve 2. Use standard deviation to mark off sections 3. Add the percentages you need
2. The Highway Patrol was conducting a study on the speed of drivers on a specific stretch of road. They found the average speed was 68 mph with a standard deviation of 5 mph. What percentage of the drivers drove between 73 to 83 mph?
Normal Approximation to a Binomial Distribution When numbers are too large to determine on a calculator or the range requires too many calculations, you may want to use the empirical rule to approximate the probability. Use the empirical rule when AND. If the sample size is large enough, these formulas allow you to approximate the normal distribution [on reference sheet] : mean = standard deviation =
3. A recent study found that 29% of adults say that they play baseball regularly. If you conduct a survey of 238 adults, what is the probability that you will find at most 55 of the adults surveyed will say they play baseball regularly? *If you tried to use the binomial formula, P(r 55) = P(0) + P(1) + P(2)... + P(55). This would be a very time consuming process!*
4. Use the normal approximation to a binomial distribution to answer the following question: If you flip a fair coin 100 times, what are the odds: a. It comes up heads 45 or less times? b. It comes up heads between 45 and 60 times? c. It comes up heads more than 55 times? Answers: For all the questions, mean = 100*.5 = 50 standard deviation = squareroot of (100*.5*.5) = 5 a. Because this is the area to the left of 1 standard deviation below the mean, you would add.15% + 2.35% + 13.5% = 16% b. Because this is the area within 1 standard deviation below the mean to 2 standard deviations above the mean, 34% + 34% + 13.5% = 81.5. c. Because this is the area more than 1 standard deviation above the mean, 13.5% + 2.35% +.15% = 16%. [Do you understand why 50% minus 34% produces the same answer?]