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Two thousand students took an exam. The scores on the exam have an approximate normal distribution with a mean μ = 81 points and standard deviation σ = 15 points.
A citrus farmer who grows mandarin oranges finds that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of 5.85 cm and a standard deviation of 0.24 cm.
a. Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.0 cm. Sketch the graph.
a. 0.2660
b. The middle 20% of mandarin oranges from this farm have diameters between ______ and ______.
b.
c. Find the 90 th percentile for the diameters of mandarin oranges, and interpret it in a complete sentence.
c. 6.16: Ninety percent of the diameter of the mandarin oranges is at most 6.15 cm.
Using the information from [link] , answer the following:
The middle area = 0.40, so each tail has an area of 0.30.
1 – 0.40 = 0.60
The tails of the graph of the normal distribution each have an area of 0.30.
Find k1 , the 30 th percentile and k2 , the 70 th percentile (0.40 + 0.30 = 0.70).
k1 = 5.72 cm
k2 = 5.98 cm
“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).
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.
Normal Distribution: X ~ N ( µ , σ ) where µ is the mean and σ is the standard deviation.
Standard Normal Distribution: Z ~ N (0, 1).
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.
Find the probability that a CD player will last between 2.8 and six years.
Find the 70 th percentile of the distribution for the time a CD player lasts.
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