6.1 The Standard Normal Distribution
- Page ID
- 36489
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Section 6.1 The Standard Normal Distribution
Learning Objective:
In this section, you will:
- Solve and explain real-world applications of probability
The standard normal distribution is a normal distribution of standardized values called z-scores. A z-score is measured in units of the standard deviation with . = 0 and . = 1. The total area under its density curve is equal to 1.
Z-Score
x−μ
If X is a normally distributed random variable and X ~ N(μ, σ), then the z-score is: z=
σ
- The z-score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ.
- Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of zero.
- The z-score allows us to compare data that are scaled differently.
Example 1: Suppose X ~ N(5, 6). This says that X is a normally distributed random variable with mean μ = 5 and standard deviation σ= 6. Suppose x= 17. Find the z-score of x.
Example 2: Suppose X ~ N(8, 1). What value of x has a z-score of -2.25?
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Example 3: Heights of women have a normal distribution with a mean of 161 cm and a standard deviation of 7 cm.
- What is the z-score for a woman who is 170 cm tall?
- How many standard deviations away from the mean is a woman who is 170cm tall?
Example 4: Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat in take and by exercising consistently. Suppose weight loss has a normal distribution. Let X = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of two pounds. X~N(5, 2).
- Suppose a person lost ten pounds in a month. Find the z-score.
- Fill in the blanks: The z-score when x = 10 pounds is z = 2.5. This z-score tells you that x = 10 is
________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
- Fill in the blanks: Suppose a person gained three pounds (a negative weight loss). Then z =
__________. This z-score tells you that x = –3 is ________ standard deviations to the __________ (right or left) of the mean.
- Suppose the random variables X and Y have the following normal distributions: X ~ N(5, 6) and Y ~ N(2, 1). If x= 17, then z= 2. (This was previously shown.) If y= 4, what is z?
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Example 5: Compare data that are scaled differently. Which is a better score? A score of 76 on a quiz where the mean score was 64 and the standard deviation was 6.7, or a score of 12 on a quiz with a mean of 10 and a standard deviation of 1.1?
Empirical Rule
If X is a random variable and has a normal distribution with mean μ and standard deviation σ, then the Empirical Rule says the following:
- About 99.7% of the x values lie between –3σ and +3σ of the mean μ Notice that almost all the x values lie within three standard deviations of the mean. The empirical rule is also known as the 68-95-99.7 rule.
Example 6: Heights of women have a normal distribution with a mean of 161 cm and a standard deviation of 7 cm.
- Approximately what is the percentage of women between 147 cm and 175 cm?
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- About 99.7% of women are between what two heights?
- What percentage of women are between 154 and 182 cm
Example 7: The mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let X = the height of a 15 to 18-year-old male from Chile in 2009 to 2010. Then X~N(170, 6.28).
- Suppose a 15 to 18-year-old male from Chile was 168cm tall from 2009 to 2010. The z-score when x = 168cm is z =_______. This z-score tells you that x = 168 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
- Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a z-score of z = 1.27. What is the male’s height? The z-score (z=1.27) tells you that the male’s height is ________ standard deviations to the __________ (right or left) of the mean.
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For more information and examples see online textbook OpenStax Introductory Statistics pages 366371.
“Introduction to Statistics” by OpenStax, used is licensed under a Creative Commons Attribution License 4.0 license