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## 5 Normal Probability Distributions

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**5 Normal Probability Distributions**Elementary Statistics Larson Farber**Section 5.1**Introduction to Normal Distributions**Properties of a Normal Distribution**x • The mean, median, and mode are equal • Bell shaped and is symmetric about the mean • The total area that lies under the curve is one or 100%**Properties of a Normal Distribution**Inflection point Inflection point x • As the curve extends farther and farther away from the mean, • it gets closer and closer to the x-axis but never touches it. • The points at which the curvature changes are called inflection points. The graph curves downward between the inflection points and curves upward past the inflection points to the left and to the right.**Empirical Rule - Review**About 95% of the area lies within 2 standard deviations About 99.7% of the area lies within 3 standard deviations of the mean About 68% of the area lies within 1 standard deviation of the mean 68%**Determining Intervals**x 3.3 3.6 3.9 4.2 4.5 4.8 5.1 An instruction manual claims that the assembly time for a product is normally distributed with a mean of 4.2 hours and standard deviation 0.3 hour. Determine the interval in which 95% of the assembly times fall. 95% of the data will fall within 2 standard deviations of the mean. 4.2 – 2 (0.3) = 3.6 and 4.2 + 2 (0.3) = 4.8. 95% of the assembly times will be between 3.6 and 4.8 hrs.**The Standard Normal Distribution**A normal distribution with a mean of 0 and a standard deviation of 1 is called the standard normal distribution. Using z-scores any normal distribution can be transformed into the standard normal distribution. z –4 –3 –2 –1 0 1 2 3 4**The Standard Score - Review**The standard score, or z-score, represents the number of standard deviations a random variable x falls from the mean.**The Standard Score**The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the standard z-score for a person with a score of: (a) 161 (b) 148 (c) 152 (a) (b) (c)**Cumulative Areas**The total area under the curve is one. z –3 –2 –1 0 1 2 3 -The cumulative area is close to 0 for z-scores close to –3.49. -The cumulative area for z = 0 is 0.5000. -The cumulative area is close to 1 for z-scores close to 3.49.**Cumulative Areas**Find the cumulative area for a z-score of –1.25. 0.1056 z –3 –2 –1 0 1 2 3 On pages A16-A17, read down the z column on the left to z = –1.2 and across to the column under .05 or The probability that z is at most –1.25 is 0.1056.**Cumulative Areas**Find the cumulative area for a z-score of –1.25. 0.1056 z –3 –2 –1 0 1 2 3 Use a graphing calculator normalcdf (-100,-1.25). The value is 0.1056, the cumulative area. The probability that z is at most –1.25 is 0.1056.**Finding Probabilities**To find the probability that z is less than a given value, read the cumulative area in the table corresponding to that z-score. Find P(z < –1.45). P (z < –1.45) = 0.0735 z –3 –2 –1 0 1 2 3 Read down the z-column to –1.4 and across to .05 or use normalcdf (-100,-1.45). The cumulative area is 0.0735.**Finding Probabilities**To find the probability that z is greater than a given value, subtract the cumulative area in the table from 1 or Use normalcdf (-1.24,100) Find P(z > –1.24). 0.1075 0.8925 z –3 –2 –1 0 1 2 3 The cumulative area (area to the left) is 0.1075. So the area to the right is 1 – 0.1075 = 0.8925. P(z > –1.24) = 0.8925**Finding Probabilities**To find the probability z is between two given values, find the areas for each and subtract the smaller area from the larger or Use normalcdf(-1.25,1.17). Find P(–1.25 < z < 1.17). z –3 –2 –1 0 1 2 3 2. P(z < –1.25) = 0.1056 1. P(z < 1.17) = 0.8790 3. P(–1.25 < z < 1.17) = 0.8790 – 0.1056 = 0.7734**Summary**To find the probability that z is less than a given value, read the corresponding cumulative area. z -3 -2 -1 0 1 2 3 To find the probability is greater than a given value, subtract the cumulative area in the table from 1. z -3 -2 -1 0 1 2 3 To find the probability z is between two given values, find the cumulative areas for each and subtract the smaller area from the larger. z 0 -3 -2 -1 1 2 3**Section 5.2**Normal Distributions Finding Probabilities**Probabilities and Normal Distributions**If a random variable x is normally distributed, the probability that x will fall within an interval is equal to the area under the curve in the interval. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the probability that a person selected at random will have an IQ score less than 115.**Probabilities and Normal Distributions**If a random variable, x is normally distributed, the probability that x will fall within an interval is equal to the area under the curve in the interval. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the probability that a person selected at random will have an IQ score less than 115. 100 115 To find the area in this interval, first find the standard score equivalent to x = 115. 115-100 15 = 1**Probabilities and Normal Distributions**Normal Distribution Find P(x < 115). 100 115 Standard Normal Distribution SAME SAME Find P(z < 1). 1 0 P(z < 1) = 0.8413, so P(x <115) = 0.8413**Application**Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected. Find the probability it is between $80 and $115.**Application**Normal Distribution Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected. Find the probability it is between $80 and $115. Using pages A16-A17 or a graphing calculator. P(80 < x < 115) = P(–1.67 < z < 1.25) =normalcdf(-1.67,1.25) = 0.8469 The probability a utility bill is between $80 and $115 is 0.8469 or 84.7%.**Section 5.3**Normal Distributions Finding Values**From Areas to z-Scores**Find the z-score corresponding to a cumulative area of 0.9803. 0.9803 –4 –3 –2 –1 0 1 2 3 4 z You can use a graphing calculator to find the z-score that corresponds to a given area. Use DISTR, invNorm or invcdf (area)**From Areas to z-Scores**Find the z-score corresponding to a cumulative area of 0.9803. z = 2.06 corresponds roughly to the 98th percentile P98. 0.9803 –4 –3 –2 –1 0 1 2 3 4 z**From Areas to z-Scores**Find the z-score corresponding to a cumulative area of 0.9803. z = 2.06 corresponds roughly to the 98th percentile. 0.9803 –4 –3 –2 –1 0 1 2 3 4 z Locate 0.9803 in the area portion of the table (A16-17). Read the values at the beginning of the corresponding row and at the top of the column. The z-score is 2.06.**Finding z-Scores from Areas**Find the z-score corresponding to the 90th percentile (P90). .90 z 0 DISTR, invNorm (.9) =**Finding z-Scores from Areas**Find the z-score corresponding to the 90th percentile (P90). .90 z 0 DISTR, invNorm (.9) = 1.28. The z score is 1.28 The closest table area is .8997. The row heading is 1.2 and column heading is .08. This corresponds to z = 1.28. A z-score of 1.28 corresponds to the 90th percentile.**Finding z-Scores from Areas**Find the z-score with an area of .60 falling to its right. .60 z 0 z**Finding z-Scores from Areas**Find the z-score with an area of .60 falling to its right. .60 z 0 z With .60 to the right, cumulative area is .40. DIST, InvNorm(.4) =**Finding z-Scores from Areas**Find the z-score with an area of .60 falling to its right. .60 z 0 z With .60 to the right, cumulative area is .40. DIST, InvNorm(.4) = -.25. The z-score is -0.25. A z-score of -0.25 has an area of .60 to its right. It also corresponds to the 40th percentile**0**Finding z-Scores from Areas Find the z-score such that 45% of the area under the curve falls between –z and z. .45 –z z**0**Finding z-Scores from Areas Find the z-score such that 45% of the area under the curve falls between –z and z. .45 –z z The area remaining in the tails is .55. Half this area is in each tail so .55/2 = .275 is the cumulative area for the -z value and .275 + .45 = .725 is the cumulative area for the positive z. Use your calculator invNorm(.275) or invNorm(.725) and the z-scores are 0.60 and -0.60.**From z-Scores to Raw Scores**To find the data value x when given a standard score z: The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the test score for a person with a standard score of: (a) 0 (b) –1.75 (c) 2.33**From z-Scores to Raw Scores**To find the data value, x when given a standard score, z: The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the test score for a person with a standard score of: (a) 2.33 (b) –1.75 (c) 0 (a) x = 152 + (0)(7) = 152 (b) x = 152 + (–1.75)(7) = 139.75 (c) x = 152 + (2.33)(7) = 168.31**Finding Percentiles or Cut-off Values**Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. What is the smallest utility bill that can be in the top 10% of the bills?**Finding Percentiles or Cut-off Values**Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. What is the smallest utility bill that can be in the top 10% of the bills? 90% 10% z Find the cumulative area in the table that is closest to 0.9000 (the 90th percentile.) The area 0.8997 corresponds to a z-score of 1.28.**Finding Percentiles or Cut-off Values**Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. What is the smallest utility bill that can be in the top 10% of the bills? $115.36 is the smallest value for the top 10%. 90% 10% z Find the cumulative area in the table that is closest to 0.9000 (the 90th percentile.) The area 0.8997 corresponds to a z-score of 1.28. To find the corresponding x-value, use x = 100 + 1.28(12) = 115.36.**Section 5.4**The Central Limit Theorem**Sampling Distributions**A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means. Sample Sample Sample Sample Sample Sample The sampling distribution consists of the values of the sample means,**The Central Limit Theorem**If a sample n 30 is taken from a population with any type distributionthat has a mean = and standard deviation = x the sample means will have a normal distribution and standard deviation**The Central Limit Theorem**If a sample of any size is taken from a population with a normal distributionwith mean = and standard deviation = x the distribution of means of sample size n, will be normal with a mean standard deviation**Application**The mean height of American men (ages 20-29) is inches. Random samples of 60 such men are selected. Find the mean and standard deviation (standard error) of the sampling distribution. mean 69.2 Distribution of means of sample size 60, will be normal. Standard deviation**Interpreting the Central Limit Theorem**The mean height of American men (ages 20-29) is = 69.2”. If a random sample of 60 men in this age group is selected, what is the probability the mean height for the sample is greater than 70”? Assume the standard deviation is 2.9”. Since n > 30 the sampling distribution of will be normal mean standard deviation Find the z-score for a sample mean of 70:**Interpreting the Central Limit Theorem**2.14 z There is a 0.0162 probability that a sample of 60 men will have a mean height greater than 70”.**Application Central Limit Theorem**During a certain week the mean price of gasoline in California was $1.164 per gallon. What is the probability that the mean price for the sample of 38 gas stations in California is between $1.169 and $1.179? Assume the standard deviation = $0.049. Since n > 30 the sampling distribution of will be normal mean standard deviation Calculate the standard z-score for sample values of $1.169 and $1.179.**Application Central Limit Theorem**P( 0.63 < z < 1.90) = 0.9713 – 0.7357 = 0.2356 z .63 1.90 The probability is 0.2356 that the mean for the sample is between $1.169 and $1.179.**Section 5.5**Normal Approximation to the Binomial**Binomial Distribution Characteristics**• There are a fixed number of independent trials. (n) • Each trial has 2 outcomes, Success or Failure. • The probability of success on a single trial is p and the probability of failure is q. p + q = 1 • We can find the probability of exactly x successes out of n trials. Where x = 0 or 1 or 2 … n. • x is a discrete random variable representing a count of the number of successes in n trials.**Application**34% of Americans have type A+ blood. If 500 Americans are sampled at random, what is the probability at least 300 have type A+ blood? Using techniques of Chapter 4 you could calculate the probability that exactly 300, exactly 301… exactly 500 Americans have A+ blood type and add the probabilities. Or…you could use the normal curve probabilities to approximate the binomial probabilities. If np 5 and nq 5, the binomial random variable x is approximately normally distributed with mean