Download Lesson 2: The Standard Normal Distribution

Lesson 2: The Standard Normal Distribution
Recall: To describe a normal distribution we use the mean and standard deviation.
Standardizing: converting the mean and standard deviation of our observations to the same units.
­ results in the same and for all normally distributed
data.
To standardize observations:
If x is an observation from a distribution that has mean and standard deviation , the standardized value of x is
This is referred to as a z­score.
A z­score tells us how many standard deviations the original observation falls away from the mean, and in which direction. Observations larger than the mean are positive when standardized, and observations smaller than the mean
are negative.
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If the variable we standardize has a normal distribution, standardizing does more than give a common scale. It makes all normal distributions into a single distribution, and this distribution is still normal. Standardizing a variable that has any normal distribution produces a new variable that has the standard normal distribution.
This standard normal distribution is the normal distribution N(0, 1) with mean 0 and standard deviation 1.
Do exercise 2.19 on page 95.
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Recall: An area under a density curve is a proportion of the observations in a distribution.
Because all normal distributions are the same when we standardize, we can find areas under any normal curve from a single table, a table that gives areas under the curve for the standard normal distribution.
Ex. 1 How to use the z­score table to find the proportion of observations from the standard normal distribution that are less than 2.34.
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Ex. 2 Find the proportion of observations from the standard normal distribution that are greater than ­2.05.
Ex. 3 What proportion of all young women's heights are less than 70 inches tall? N(64.5, 2.5)
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FINDING NORMAL PROPORTIONS
Step 1: State the problem in terms of the observed variable x. Draw a picture of the distribution and shade the area of interest under the curve.
Step 2: Standardize x to restate the problem in terms of a standard normal variable z. Draw a picture to show the area of interest under the standard normal curve.
Step 3: Find the required area under the standard normal curve, using Table A and the fact that the total area under the curve is 1.
Step 4: Write your conclusion in the context of the problem.
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Ex. 4 What percent of 14­year­old boys have blood cholesterol between 150 and 230 mg/dl? The variable x has the N(170, 30) distribution.
Do exercises 2.22 ­ 2.25 on p. 103
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Finding a value given a proportion
Ex. 4
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