CHAPTER 6 The Normal Distribution Student Notes Stat 200 Elementary Statistics Introduction Many continuous variables have distributions that are bell shaped and are called approximately normally distributed variables normal distribution is also known as the bell curve or the Gaussian distribution A Normal Distribution Symmetrical Skewed Section 6 1 Probabilities of a Normal Distribution Formula Graph Standard Norm al Distribution y e X m 2 2 2s 0 5 0 4 s 2p 0 3 0 2 0 1 0 4 3 2 1 0 z score 1 2 3 4 Shape and position of the Normal Distribution Curve 1 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 Distribution Standard Normal DistributionZ values z Finding Area Under the St Normal Dist 1 2 value mean X or z standard deviation 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 b P 0 z 2 32 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 Distribution Example 1 The 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 2 Each 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 b More 27 and 31 pounds per month than 30 2 Example 3 To 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 X X z Example 4 For 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 the study Determining Normality Checking normality Pearson s Index PI 3 X median s Example 5 A 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 74 81 88 91 97 98 113 118 151 158 Example 6 The 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 70 6 3 The Central Limit Theorem Sampling Distribution of Sample Means is a distribution obtained by using the means computed from 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 standard error of the mean and is called the 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 1 A 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 2 The 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 3 The 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 that the mean of the sample will be less than 224 ponds per year Finite Population Correction Factor 6 4 The Normal Approximation to the Binomial Distribution Correction for continuity Characteristics of a Binomial Distribution There must be a fixed number of trials The outcome of each trial must be independent Each experiment can have only two outcomes or be reduced to two outcomes The probability of a success must remain the same for each …