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7.2 Using the normal distribution  (Page 4/25)

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References

“Naegele’s rule.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Naegele's_rule (accessed May 14, 2013).

“403: NUMMI.” Chicago Public Media&Ira Glass, 2013. Available online at http://www.thisamericanlife.org/radio-archives/episode/403/nummi (accessed May 14, 2013).

“Scratch-Off Lottery Ticket Playing Tips.” WinAtTheLottery.com, 2013. Available online at http://www.winatthelottery.com/public/department40.cfm (accessed May 14, 2013).

“Smart Phone Users, By The Numbers.” Visual.ly, 2013. Available online at http://visual.ly/smart-phone-users-numbers (accessed May 14, 2013).

“Facebook Statistics.” Statistics Brain. Available online at http://www.statisticbrain.com/facebook-statistics/(accessed May 14, 2013).

Chapter review

The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean µ and the standard deviation σ . A special normal distribution, called the standard normal distribution is the distribution of z -scores. Its mean is zero, and its standard deviation is one.

Formula review

Normal Distribution: X ~ N ( µ , σ ) where µ is the mean and σ is the standard deviation.

Standard Normal Distribution: Z ~ N (0, 1).

Calculator function for probability: normalcdf (lower x value of the area, upper x value of the area, mean, standard deviation)

Calculator function for the k th percentile: k = invNorm (area to the left of k , mean, standard deviation)

How would you represent the area to the left of one in a probability statement?

P ( x <1)

What is the area to the right of one?

Is P ( x <1) equal to P ( x ≤ 1)? Why?

Yes, because they are the same in a continuous distribution: P ( x = 1) = 0

How would you represent the area to the left of three in a probability statement?

What is the area to the right of three?

1 – P ( x <3) or P ( x >3)

If the area to the left of x in a normal distribution is 0.123, what is the area to the right of x ?

If the area to the right of x in a normal distribution is 0.543, what is the area to the left of x ?

1 – 0.543 = 0.457

Use the following information to answer the next four exercises:

X ~ N (54, 8)

Find the probability that x >56.

Find the probability that x <30.

0.0013

Find the 80 th percentile.

Find the 60 th percentile.

56.03

X ~ N (6, 2)

Find the probability that x is between three and nine.

X ~ N (–3, 4)

Find the probability that x is between one and four.

0.1186

X ~ N (4, 5)

Find the maximum of x in the bottom quartile.

Use the following information to answer the next three exercise: The life of Sunshine CD players is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts. Find the probability that a CD player will break down during the guarantee period.

  1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
    Empty normal distribution curve.
  2. P (0< x <____________) = ___________ (Use zero for the minimum value of x .)
  1. Check student’s solution.
  2. 3, 0.1979

Find the probability that a CD player will last between 2.8 and six years.

  1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
    Empty normal distribution curve.
  2. P (__________< x <__________) = __________

Find the 70 th percentile of the distribution for the time a CD player lasts.

  1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.
    Empty normal distribution curve.
  2. P ( x < k ) = __________ Therefore, k = _________
  1. Check student’s solution.
  2. 0.70, 4.78 years

Questions & Answers

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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
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Prevent foreign microbes to the host
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Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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