CHAPTER 6 The Normal DistributionStat 200 – Elementary StatisticsIntroductionSection 6.1 Probabilities of a Normal DistributionStandard Normal DistributionStat 200 – Elementary Statistics IntroductionMany continuous variables have distributions that are bell-shapedand are called approximately normally distributed variables.A normal distribution is also known as the bell curve or the Gaussian distribution.Normal DistributionSymmetricalSkewedSection 6.1 Probabilities of a Normal DistributionFormula GraphStudent NotesCHAPTER 6 The Normal Distribution2 2( ) (2 )2Xeym ss p- -=Standard Norm al Distribution00.10.20.30.40.5-4 -3 -2 -1 0 1 2 3 4z score Shape and position of the Normal Distribution Curve1. The normal distribution curve is _________________.2. The mean, median, and mode are ________ and located at the _________ of the distribution.3. The normal distribution curve is _____________(i.e., it has only one mode).4. The curve is ______________about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.5. The curve is ________________—i.e., there are no gaps or holes.6. The curve never touches the ___________. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly closer.7. The total area under the normal distribution curve is equal to ___________________. 8. The ________ under the normal curve that lies within one standard deviation of the mean is approximately 0.68, or 68%; within two standard deviations, about 0.95, or 95%; and within three standard deviations, about 0.997 or 99.7%. Standard Normal DistributionStandard Normal Distribution- Z valuesFinding Area Under the St.Normal Dist1.2.z zXvalue meanstandard deviation or 3.Find the area with a score:#1 a. between –2.48 and –0.83 b. Between z = +1.68 and z = -1.37 c. To the left of z = 1.99#2 a. between 0 and 1.58 b. greater than +1.27 c. between 1.2 and 2.3#3 Find the area to the right of z = +2.43 and to the left of z = -3.01#4 a. P(0  z  2.32) b. P(z < 1.65) c. P(z > 1.91) Find the z value such that the area under the standard normal distribution curve between 0 and the z value is 0.2123.6.2 Applications of the Normal DistributionExample 1The mean number of hours an American worker spends on a computer is 3.1 hours per workday. Assume the standard deviation is 0.5 hours. Find the percentage of workers who spend less than 3.5 hours on the computer. Assume the variable is normally distributed.Example 2Each month, an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the standard deviation is 2 pounds. If a household is selected at random, find the probability of its generating:a.Between 27 and 31 pounds per month.b.More than 30.2Example 3To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed.Finding XExample 4For a medical study, a researcher wishes to select people in the middle 60% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify people to participate in thestudy.Determining NormalityChecking normalityPearson’s IndexsmedianXPI)(3 zXExample 5A survey of 18 high-technology firms showed the number of day’s inventory they had on hand. Determine if the data are approximately normally distributed.5 29 34 44 45 63 68 74 7481 88 91 97 98 113 118 151 158Example 6The data shown consist of the number of games played each year in the career of Baseball Hall of Famer Bill Mazeroski. Determine if the data are approximately normally distributed.81 148 152 135 151 152 159 142 34 162 130 162 163 143 67 112 706.3 The Central Limit TheoremSampling Distribution of Sample Means is a distribution obtained by using the means computedfrom random samples of a specific size taken from a population.Sampling Error - is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.Properties of Distribution of Sample Means The mean of the sample means will be the same as the population mean. The standard deviation of the sample means will be smaller than the standard deviation of the population, and will be equal to the population standard deviation divided by the square root of the sample size.The Central Limit Theorem As the sample size n increases, the shape of the distribution of the sample means taken with replacement from a population with mean m and standard deviation s will approach a normal distribution.  If all possible samples of size n are taken with replacement from the same population, the mean of the sample means equals the population mean or: . The standard deviation of the sample means equals: and is called the standard error of the mean. The central limit theorem can be used to answer questions about sample means in the same manner that the normal distribution can be used to answer questions about individual values. A new formula must be used for the z values:Example 1A.C. Nelson reported that children between the ages of 2 and 5 watch an average of 25 hours of television per week. Assume the variable in normally distributed and the standard deviation is 3 hours. If 20 children between the ages of 2 and 5 are randomly selected, find the probability that the mean of the number f hours they watch television will be greater than 26.3 hours.Example 2The average age of a vehicle registered in the United States is 8 years, or 96 months. Assume the standard deviation is 16 months. If a random sample of 36 vehicles is selected, find the probability that the mean of their age is between 90 and 100 months.Example 3The average number of pounds of meat that a person consumes a year is 218.4 pounds. Assume that the standard deviation is 25 pounds and the distribution is approximately normal.a. Find the probability that a person selected at random consumes less than 224 pounds per year.b. If a sample of 40 individuals is selected, find the probability