BCOR 1020 Business Statistics

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BCOR 1020Business Statistics

• Lecture 13 February 28, 2008

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Overview

• Chapter 7 Continuous Distributions
• Normal Distribution (continued)

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Chapter 7 Normal Distribution
Recall the Standard Normal

• Since for every value of m and s, there is a
  different normal distribution, we transform a
  normal random variable to a standard normal
  distribution with m 0 and s 1 using the
  formula

• Shift the point of symmetry to zero by
  subtracting m from x.
• Divide by s to scale the distribution to a normal
  with s 1.

• Denoted N(0,1)
• Appendix C-2 allows you to find all of the area
  under the curve left of z. (Hand-out)

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Chapter 7 Normal Distribution

• Example Using the Std. Normal transformation
• Daily sales at a bicycle shop are normally
  distributed with mean 15,000 and standard
  deviation 4000.
• Find the probability that sales will exceed
  20,000 on a randomly-selected day.
• Find the probability that sales will be less than
  12,000 on a randomly selected day.
• Find the probability that sales will be between
  12,000 and 20,000 on a randomly selected day.

P(X gt 20000) 1 P(X lt 20000)
P(X lt 12000)
P(12000 lt X lt 20000) P (X lt 20000) P(X lt
12000)
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Clickers
If the starting salary for students majoring in
Business is normally distributed with a mean of
45,000 and a standard deviation of 5,000, find
the probability that the starting salary of a
randomly selected student will be less than
50,000. A 0.1056 B 0.3085 C
0.6915 D 0.8413
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Clickers
If the starting salary for students majoring in
Business is normally distributed with a mean of
45,000 and a standard deviation of 5,000, find
the probability that the starting salary of a
randomly selected student will be at least
50,000. A 0.1056 B 0.1587 C
0.8413 D 0.8944
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Clickers
If the starting salary for students majoring in
Business is normally distributed with a mean of
45,000 and a standard deviation of 5,000, find
the probability that the starting salary of a
randomly selected student will be between
45,000 and 50,000. A 0.3413 B
0.3944 C 0.8413 D 0.8944
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Chapter 7 Normal Distribution
Basis for the Empirical Rule

• Approximately 68 of the area under the curve is
  between 1s.
• Approximately 95 of the area under the curve is
  between 2s.
• Approximately 99.7 of the area under the curve
  is between 3s.

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Chapter 7 Normal Distribution
Finding z for a Given Area

• Appendices C-1 and C-2 be used to find the
  z-value corresponding to a given probability.
• For example, what z-value defines the top 1 of a
  normal distribution?
• This implies that 99 of the area lies to the
  left of z.
• Or that 1 of the area lies to the left of z.
• Or that 49 of the area lies between 0 and z.

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Chapter 7 Normal Distribution
Finding z for a Given Area

• Look for an area of .4900 in Appendix C-1 (or
  for an area of 0.9900 in the Appendix C-2 the
  handout)

• Without interpolation, the closest we can get is
  z 2.33

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Chapter 7 Normal Distribution
Finding z for a Given Area

• Some important Normal areas

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Chapter 7 Normal Distribution
Finding Areas by Standardizing

• Suppose John took an economics exam on which the
  class mean was 75 with a standard deviation of 7.
  What score would place John in the upper 10th
  percentile?

• From the previous slide, we know P(Z gt 1.282)
  .10.

• Find the value of x such that P(X gt x) .10 or
  P(X lt x) 0.90.

• Since

• A score of 84 or better would place John in the
  top 10 of his class.

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Clickers
Suppose the starting salary for students majoring
in Business is normally distributed with a mean
of 45,000 and a standard deviation of 5,000.
If Jane Wants a starting salary in the top 25,
approximately what salary should she negotiate
for? A 56,630 B 54,800 C
53,225 D 51,410 E 48,375
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Chapter 7 Normal Distribution

• Normal Probabilities and z-scores in MegaStat
  (time allowing)