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Section 8.5 Normal Distributions Density curve: the smooth curve identifying the shape of a distribution. Basic Properties of Density Curves Property 1: A density curve is always on or above the horizontal axis. Property 2: The total area under a density curve (and above the horizontal axis) equals 1 Variables and Their Density Curves For a variable with a density curve, the percentage of all possible observations of the variable that lie within any specified range equals (at least approximately) the corresponding area under the density curve, expressed as a percentage. 60% of the data is within the values of A and B 0.2 1 0.1 A B This section will focus on the most important density curve-the normal density curve. The normal density curve has a symmetric “bell-shaped” curve Normal distribution: is a distribution represented by a normal curve. What is important about the normal distribution? It can be used as a more convenient approximation for a variable X which we will be exploring in this section How can we express the normal curve as a distribution for X? 1. As a smooth approximation to a histogram that is based upon a sample of X. 2. As an idealized representation for the population distribution of X by knowing µ-the population mean and σthe population standard deviation. If a variable X follows normal distribution, then it can be represented by X ~ N , . N represents that the distribution is normally distributed, represents the population mean, and represents the population standard deviation. All normal curves can be described by this single formula. If a variable X follows a normal distribution with mean and standard deviation , then the density curve of the distribution of X is given by the formula f ( x) 1 e 2 1 x 2 2 Where f (x) is called the density function e is a constant about 2.71 is a constant about 3.14 µ-3σ µ-2σ µ-σ µ µ+σ µ+2σ µ+3σ Example 1 Sketch the normal distribution with a. µ = -2 and σ = 2 b. µ = -2 and σ = 1/2 c. µ = 0 and σ = 2 Can we look at these distributions in our calculator? Yes Graphing Procedure 1 1. In your graphing mode go to 2. Y1= 3. Press 2 VARS →DISTR 4. You will see DISTR DRAW 5. Press 1:normalpdf( 6. Press x,-2, 2 ) You will see the first distribution. You may need to adjust your window the y-values will not be negative and the maximum y-value will not be larger than 1 for these graphs We looked at relative frequency histograms and density curves to find the probabilities that our X-values were between a certain range. Now we will be applying that same sort of idea with our normal curve. Normally Distributed Variables and Normal-Curves Areas For a normally distributed variable, the percentage of all possible observations that lie within any specified range equals the corresponding area under its associated normal curve, expressed as a percentage. This result holds approximately for a variable that is approximately normally distributed. Example 2 The area under a particular normal curve between 10 and 15 is 0.6874. A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie between 10 and 15? Explain your answer. The most common normal curve X ~ N , is the standard normal curve Z ~ N 0,1 There are tables made up so we can find the area under curve if we have a z-table. Standard Normal distribution: Standard Normal Curve A normally distributed variable having mean 0 and standard deviation 1 is said to have the standard normal distribution. Its associated normal curve is called the standard normal curve. Basic Properties of the Standard Normal Curve Property 1: The total area under the standard normal curve is 1. Property 2: The standard normal curve extends indefinitely in both directions, approaching but never touching, the horizontal axis as it does so. Property 3: The standard normal curve is symmetric about 0; that is, the part of the curve to the left of the dashed line is the mirror image of the part of the curve to the right of it. Property 4: Almost all the area under the standard normal curve lies between -3 and 3. -3 -2 -1 0 1 2 3 Procedure 1: Expressing the range in terms of z-scores and finding the corresponding area. Step 1: Draw a standard normal curve. Step 2: Label the z-score(s) on the curve. Step 3: Shade in the region of interest. Step 4: Determining the corresponding area under the standard normal curve using table in Appendix B (pg A18). -3 -2 -1 0 1 2 3 Example 3 Z is the standard normal distribution. Find the indicated probabilities. a. P0 Z 1.5 b. P 1.3 Z 0 c. P 1.64 Z 2.35 d. PZ 1.82 e. P 2.6 Z Can we do this on our calculator? Sort of, but we need to draw our picture(s) so we make sure that we know what area we are interested in Graphing Procedure 2 Method to find some of the areas in our calculator 1. Press 2 VARS →DISTR 2. You will see DISTR DRAW 3. Press 2:normalcdf( 4. Enter in first value , Enter in second value ) 5. Press Enter We can standardize any normally distributed variable(s) so we can find the area under the curve Standardized Normally Distributed Variable Z The standardized version of a normally distributed variable X , x has the standard normal distribution. Z X µ-3σ -3 µ-2σ µ-σ -2 -1 µ 0 µ+ σ 1 µ+2σ µ+3σ Z 2 3 Procedure 2: To Determine a Percentage or Probability for a Normally Distributed Variable Step 1: Sketch the normal curve associated with the variable Step 2: Shade the region of interest and mark its delimiting xvalue(s). Step 3: Find the z-score(s) for the delimiting x-value(s) found in Step 2. Step 4: Use the standard normal curve in Appendix B to find the area under the standard normal curve delimited by the zscore(s) found in Step 3. Example 4 X has a normal distribution with the given mean and standard deviation. Find the indicated probabilities. 9. 50 , 10 , find P(35 X 65) 14. 100 , 15 , find P(70 X 80) Can we do this on our calculator? Sort of, but we need to draw our picture(s) so we make sure that we know what area we are interested in Graphing Procedure 3 Method to find some of the probabilities in our calculator if we know the X values, , and . 1. Press 2 VARS →DISTR 2. You will see DISTR DRAW 3. Press 2:normalcdf( 4. Enter in first value , Enter in second value, type in , ) 5. Press Enter Example 5 20. X has a normal random variable with mean 10 and standard deviation 5 . Find b such that P(10 X b) 0.4 27. SAT scores are normally distributed with a mean 500 and a standard deviation of 100. Find the probability that a randomly chosen test-taker will score between 450 and 550. 30. LSAT test scores are normally distributed with a mean of 151 and a standard deviation of 7. Find the probability that a randomly chosen test taker will score 144 or lower. 34. If the mean IQ score is 100 and standard deviation is 16 find the number of people in the U.S. if the population is 313,000,000 with an IQ of 140 or higher. 36. LSAT test scores are normally distributed with a mean of 151 and a standard deviation of 7.What score would place you in the top 2 of test takers? Empirical Rule For any data set having roughly a bell-shaped distribution. Approximately 68% of the observations lie within one standard deviation to either side of the mean. This would , for populations. Approximately 95% of the observations lie within two standard deviations to either side of the mean. This would 2 , 2 for populations. Approximately 99.7% of the observations lie within three standard deviations to either side of the mean. This would 3 , 3 for populations.