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StatisticsNormal Probability Distributions
Chapter 6Example Problems
Normal Probability Distributions
• Mean = 0• Standard Deviation = 1
Normal Probability Distribution
• Table in back of textbook– Negative z scores– Positive z scores
Normal Probability Distribution
• Find the area of the shaded region
• Look in table and find row 0.6 and column 0.06 as this is 0.66
z = 0.66
Normal Probability Distribution
• Question: Assume the readings on thermometers are normally distributed with a mean of 0o C and a standard deviation of 1.00o C.
• Find the probability that a randomly selected thermometer reads greater than 2.26 and draw a sketch of the region.
Normal Probability Distribution
• Answer: We know that we are going to shade to the right of our given value of 2.26 because it said "greater than". So we need to find this z value
z = (x - mean) / standard deviation z = (2.26-0)/1 = 2.26
– So we look up in our table the value of z = 2.26. I look in the Positive z Scores table and look in the row 2.2 and column .06 and find 0.9881. • But remember this is not the area to the right of 2.26, this
would be the area to the left of 2.26. So I need to take 1 - 0.9881 (remember the entire area adds to 1) = 0.0119.
Calculator
• If you want to use your calculator to find the z score instead of the table,
• 2nd VARS• normalcdf(2.26, 10)• NOTE: The 10 is just giving a very large value
because we cannot tell the calculator where to stop or to go to infinity.
Normal Probability Distributions
• Question: Assume that women's heights are normally distributed (this is the key to use the normal z table) with a mean given by mean = 63.4 inches and a standard deviation = 1.8 inches.
• If a woman is randomly selected, find the probability that her height is between 62.9 inches and 63.9 inches.
Normal Probability Distributions
• Answer: Find the z value with the formula z = (x – mean) / standard deviation for the lower value of x = 62.9 and upper value x = 63.9
Low z = (62.9 – 63.4) / 1.8 = -0.28 (rounded to two decimals)Upper z = (63.9 – 63.4) / 1.8 = 0.28 (rounded to two decimal places)
• So “between” would be the area between z = -0.28 and z = 0.28
Normal Probability Distributions
• The area from the bottom of the normal curve up to z = -0.28 is found in the Negative z scores table. Look up -0.2 in the row and .08 in the column to get .3897.
• The area from the bottom of the normal curve up to z = 0.28 is found in the Positive z scores table. Look up 0.2 in the row and .08 in the column to get .6103.– Since this is the area between, if you subtract these two
values you get the area between• .6103 - .3897 = .2206
