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5.1 The standard normal distribution  (Page 2/14)

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Fill in the blanks.

Jerome averages 16 points a game with a standard deviation of four points. X ~ N (16,4). Suppose Jerome scores ten points in a game. The z –score when x = 10 is –1.5. This score tells you that x = 10 is _____ standard deviations to the ______(right or left) of the mean______(What is the mean?).

1.5, left, 16

The 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 68% of the x values lie between –1 σ and +1 σ of the mean µ (within one standard deviation of the mean).
  • About 95% of the x values lie between –2 σ and +2 σ of the mean µ (within two standard deviations of the mean).
  • About 99.7% of the x values lie between –3 σ and +3 σ of the mean µ (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.
  • The z -scores for +1 σ and –1 σ are +1 and –1, respectively.
  • The z -scores for +2 σ and –2 σ are +2 and –2, respectively.
  • The z -scores for +3 σ and –3 σ are +3 and –3 respectively.

The empirical rule is also known as the 68-95-99.7 rule.

This frequency curve illustrates the empirical rule. The normal curve is shown over a horizontal axis. The axis is labeled with points -3s, -2s, -1s, m, 1s, 2s, 3s. Vertical lines connect the axis to the curve at each labeled point. The peak of the curve aligns with the point m.

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).

a. Suppose a 15 to 18-year-old male from Chile was 168 cm tall from 2009 to 2010. The z -score when x = 168 cm 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?).

a. –0.32, 0.32, left, 170

b. 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.

b. 177.98, 1.27, right

Try it

Use the information in [link] to answer the following questions.

  1. Suppose a 15 to 18-year-old male from Chile was 176 cm tall from 2009 to 2010. The z -score when x = 176 cm is z = _______. This z -score tells you that x = 176 cm is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
  2. Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a z -score of z = –2. What is the male’s height? The z -score ( z = –2) tells you that the male’s height is ________ standard deviations to the __________ (right or left) of the mean.

Try it solutions

Solve the equation z = x − μ σ for x . x = μ + ( z )( σ )

  1. z = 176 − 170 6.28 ≈ 0.96, This z -score tells you that x = 176 cm is 0.96 standard deviations to the right of the mean 170 cm.
  2. X = 157.44 cm, The z -score( z = –2) tells you that the male’s height is two standard deviations to the left of the mean.

From 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let Y = the height of 15 to 18-year-old males from 1984 to 1985. Then Y ~ N (172.36, 6.34).

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).

Find the z -scores for x = 160.58 cm and y = 162.85 cm. Interpret each z -score. What can you say about x = 160.58 cm and y = 162.85 cm?

The z -score for x = 160.58 is z = –1.5.
The z -score for y = 162.85 is z = –1.5.
Both x = 160.58 and y = 162.85 deviate the same number of standard deviations from their respective means and in the same direction.
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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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