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Lecture 22 Eng. Ismail Zakaria El Daour 2010. Probabilistic and Statistical Techniques. Chapter 6 (part 3) Normal Probability Distribution. Probabilistic and Statistical Techniques. Probabilistic and Statistical Techniques. Table A-2. Converting to a Standard Normal Distribution.
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Lecture 22Eng. Ismail Zakaria El Daour 2010 Probabilistic and Statistical Techniques
Chapter 6 (part 3) Normal Probability Distribution Probabilistic and Statistical Techniques
Probabilistic and Statistical Techniques Table A-2
Converting to a Standard Normal Distribution x – z = Probabilistic and Statistical Techniques
Probabilistic and Statistical Techniques In a specific case, we noted that the safe load for a water taxi was found to be 3500 pounds. We also noted that the mean weight of a passenger was assumed to be 140 pounds. Assume the case that all passengers are men. Assume also that the weights of the men are normally distributed with a mean of 172 pounds and standard deviation of 29 pounds. If one man is randomly selected, what is the probability he weighs less than 174 pounds? Example Weights of Water Taxi Passengers
Example - cont = 172 =29 174 – 172 z = = 0.07 29 z=0 z= 0.07 Probabilistic and Statistical Techniques
Example - cont = 172 =29 Probabilistic and Statistical Techniques P ( x < 174 lb.) = P(z < 0.07) = 0.5279
Probabilistic and Statistical Techniques • Many processes such as filling soda bottles and canning fruit, are normally distributed. Manufacturers must guard against both over and underfilling. If they put too much in the can or bottle, they are giving away their product. If they put too little in, the customer may fell cheated and the government may question the label description. • “Control Chart” with limits drawn three standard deviations above and below the mean, are routinely used to monitor this type of production process. Control Charts
Probabilistic and Statistical Techniques Domino Sugar packets are labeled as containing 3.5g. Assume those packets are actually filled with amounts that are normally distributed with a mean of 3.586 g. and a standard deviation of 0.074 g. What percentage of packets have less than 3.5 g.? Are many consumers being cheated? Example
Probabilistic and Statistical Techniques Weights of newborn babies in U.S.A are normally distributed with a mean of 3420g and a standard deviation of 495g. A newborn baby weighing less than 2200g is considered to be at risk. a) What percentage of newborn babies are ‘at risk’ category? b) If Chicago General Hospital has 900 births in a year, how many of the babies are in the ‘at risk’ category? Example
Probabilistic and Statistical Techniques • The mean starting salaries for college graduates in the spring of 2004 was 36,280$. Assume that the distribution of the starting salaries of college graduates follows normal distribution with a standard deviation of 3,300$. What percent of graduates have starting salaries: • between 35,000 and 40,000$ • more than 45,000$ • between 40,000 and 45,000$ Example
Probabilistic and Statistical Techniques • 1. Don’t confuse z scores and areas. z scores are distances along the horizontal scale, but areas are regions under the normal curve. Table A-2 lists z scores in the left column and across the top row, but areas are found in the body of the table. • 2. Choose the correct (right/left) side of the graph. • 3. A z score must be negative whenever it is located in the left half of the normal distribution. • 4. Areas (or probabilities) are positive or zero values, but they are never negative. Cautions to Keep in Mind
Probabilistic and Statistical Techniques 1.Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the x value(s) being sought. 2.Use Table A-2 to find the z score corresponding to the cumulative left area bounded by x. Refer to the body of Table A-2 to find the closest area, then identify the corresponding z score. Procedure for Finding Values Using Table A-2
Probabilistic and Statistical Techniques • Using the following Formula, enter the values for µ, , and the z score found in step 2, then solve for x. • x = µ + (z • ) • (If z is located to the left of the mean, be sure that it is a negative number.) • 4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem. Procedure for Finding Values Using Table A-2
Example – Lightest and Heaviest - cont = 172 =29 Probabilistic and Statistical Techniques Use the data from the previous example to determine what weight separates the lightest 99.5% from the heaviest 0.5%?
Probabilistic and Statistical Techniques Example – Lightest and Heaviest - cont x = + (z● ) x = 172 + (2.575 29) x = 246.675 (247 rounded)
Example – Lightest and Heaviest - cont Probabilistic and Statistical Techniques The weight of 247 pounds separates the lightest 99.5% from the heaviest 0.5%
Probabilistic and Statistical Techniques Assume that women heights are normally distributed with a mean of 63.6 in and a standard deviation of 2.5 in. To be eligible for the U.S. army, a women height must be between 58 and 80 in. if the requirement is changed so that only the shortest 1% and tallest 1% are excluded, find the minimum and maximum acceptable heights Example
