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A density curve is the graph of a continuous probability distribution.
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Basic Properties of the Standard Normal Curve
1. The total area under the standard normal curve is 1
2. The standard normal curves extends indefinitely in both directions, approaching, but never touching , the horizontal axis.
3. The standard normal curve is symmetric about 0.
4. Almost all the area under the standard normal curve lies between -3 and 3.
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Areas under the standard normal curve
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Because the total area under the densitycurve is equal to 1, there is a correspondence between area and probability
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If thermometers have an average (mean) reading of 0degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that, at the freezing point of water, the reading is less than 1.58 degrees
94.29% of the thermometers have readings less than 1.58 degrees.
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94.29% of the thermometers have readings less than 1.58 degrees.
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xz
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If Z is a standard normal variable, find the probability that Z lies between 0.7 and 1.98.
A. 0.2175
B. -0.2181
C. 1.7341
D. 0.2181
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If Z is a standard normal variable, find the probability that Z lies between 0.7 and 1.98.
A. 0.2175
B. -0.2181
C. 1.7341
D. 0.2181
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Finding z Scores When Given Probabilities
(z score will be positive)
Z = 1.645
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Converting to a Standard Normal Distribution
x – z =
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The safe load for a water taxi was found to be 3500 pounds. 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?
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z = 174 – 172
29 = 0.07 =29
= 172
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P ( x < 174 lb.) = P(z < 0.07) = 0.5279
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Using the calculator
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120 110.83
12z
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