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 80 th percentile.
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.
- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
- P (0< x <____________) = ___________ (Use zero for the minimum value of x .)
- Check student’s solution.
- 3, 0.1979
Find the probability that a CD player will last between 2.8 and six years.
- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
- P (__________< x <__________) = __________
Find the 70 th percentile of the distribution for the time a CD player lasts.
- Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.
- P ( x < k ) = __________ Therefore, k = _________
- Check student’s solution.
- 0.70, 4.78 years