Statistics
Normal Probability Distributions
Chapter 6
Example 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