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3/14/2013 7-1: Graphs of Normal Probability Distributions Important properties of a normal curve: 1. The curve is bell-shaped with the highest point over the mean 2. The curve is symmetrical about a vertical line through the mean 3. The curve approaches the horizontal axis but never touches or crosses it 4. The inflection (transition) points between cupping upward and downward occur above and Diagram: 1 3/14/2013 Two Normal Curves Both curves have the same mean, µ = 6. Curve A has a standard deviation of σ = 1. Curve B has a standard deviation of σ = 3. Normal Probability • The area under any normal curve will always be 1. • The portion of the area under the curve within a given interval represents the probability that a measurement will lie in that interval. 2 3/14/2013 Empirical rule: For a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution): • Approximately 68% of the data values will lie within one standard deviation on each side of the mean. • Approximately 95% of the data values will lie within two standard deviations on each side of the mean. • Approximately 99.7% (or almost all) of the data values will lie within three standard deviations on each side of the mean. Empirical Rule diagram: 3 3/14/2013 Empirical Rule Example • Suppose that wheat yield per acre is normally distributed with a mean of 35 bushels and a standard deviation of 8 bushels. • Graph the distribution. • Find the probability that an acre will yield between 19 and 35 bushels. Empirical Rule Example 4 3/14/2013 Empirical Rule Example Examples: A) Sketch a normal curve with a mean of 10 and standard deviation 3. 5 3/14/2013 B) The time required to complete today’s homework assignment is normally distributed with a mean of 30 minutes and standard deviation of 4 minutes. a) What percentage of students will take between 22 and 38 minutes? b) If there are 20 students, how many will take less than 26 minutes? c) Estimate the range of time centered about the mean in which about 68% of the students’ homework time will be contained. d) Estimate the probability that the assignment will not take longer than 34 minutes. ASSIGNMENT: p. 253-255 #1-11 (4- a,b,e only) 6