The%20Mean%20of%20a%20Discrete%20Probability%20Distribution

1
The Mean of a Discrete Probability Distribution

• The mean of a probability distribution for a
  discrete random variable is
• where the sum is taken over all possible values
  of x.

2
Which Wager do You Prefer?

• You are given 100 and told that you must pick
  one of two wagers, for an outcome based on
  flipping a coin
• A. You win 200 if it comes up heads and lose
  50 if it comes up tails.
• B. You win 350 if it comes up head and lose
  your original 100 if it comes up tails.
• Without doing any calculation, which wager would
  you prefer?

3
You win 200 if it comes up heads and lose 50 if
it comes up tails.

• Find the expected outcome for this
• wager.
• 100
• 25
• 50
• 75

4
You win 350 if it comes up head and lose your
original 100 if it comes up tails.

• Find the expected outcome for this
• wager.
• 100
• 125
• 350
• 275

5
Section 6.2

• How Can We Find Probabilities for Bell-Shaped
  Distributions?

6
Normal Distribution

• The normal distribution is symmetric, bell-shaped
  and characterized by its mean µ and standard
  deviation s.
• The probability of falling within any particular
  number of standard deviations of µ is the same
  for all normal distributions.

7
Normal Distribution
8
Z-Score

• Recall The z-score for an observation is the
  number of standard deviations that it falls from
  the mean.

9
Z-Score

• For each fixed number z, the probability within z
  standard deviations of the mean is the area under
  the normal curve between

10
Z-Score

• For z 1
• 68 of the area (probability) of a normal
• distribution falls between

11
Z-Score

• For z 2
• 95 of the area (probability) of a normal
• distribution falls between

12
Z-Score

• For z 3
• Nearly 100 of the area (probability) of a normal
• distribution falls between

13
The Normal Distribution The Most Important One
in Statistics

• Its important because
• Many variables have approximate normal
  distributions.
• Its used to approximate many discrete
  distributions.
• Many statistical methods use the normal
  distribution even when the data are not
  bell-shaped.

14
Finding Normal Probabilities for Various Z-values

• Suppose we wish to find the probability within,
  say, 1.43 standard deviations of µ.

15
Z-Scores and the Standard Normal Distribution

• When a random variable has a normal distribution
  and its values are converted to z-scores by
  subtracting the mean and dividing by the standard
  deviation, the z-scores have the standard normal
  distribution.

16
Example Find the probability within 1.43
standard deviations of µ
17
Example Find the probability within 1.43
standard deviations of µ

• Probability below 1.43s .9236
• Probability above 1.43s .0764
• By symmetry, probability below -1.43s .0764
• Total probability under the curve 1

18
Example Find the probability within 1.43
standard deviations of µ
19
Example Find the probability within 1.43
standard deviations of µ

• The probability falling within 1.43 standard
  deviations of the mean equals
• 1 0.1528 0.8472, about 85

20
How Can We Find the Value of z for a Certain
Cumulative Probability?

• Example Find the value of z for a cumulative
  probability of 0.025.

21
Example Find the Value of z For a Cumulative
Probability of 0.025
Example Find the Value of z For a Cumulative
Probability of 0.025

• Look up the cumulative probability of 0.025 in
  the body of Table A.
• A cumulative probability of 0.025 corresponds to
  z -1.96.
• So, a probability of 0.025 lies below µ - 1.96s.

22
Example Find the Value of z For a Cumulative
Probability of 0.025
23
Example What IQ Do You Need to Get Into Mensa?

• Mensa is a society of high-IQ people whose
  members have a score on an IQ test at the 98th
  percentile or higher.

24
Example What IQ Do You Need to Get Into Mensa?

• How many standard deviations above the mean is
  the 98th percentile?
• The cumulative probability of 0.980 in the body
  of Table A corresponds to z 2.05.
• The 98th percentile is 2.05 standard deviations
  above µ.

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Example What IQ Do You Need to Get Into Mensa?

• What is the IQ for that percentile?
• Since µ 100 and s 16, the 98th percentile of IQ
  equals
• µ 2.05s 100 2.05(16) 133

26
Z-Score for a Value of a Random Variable

• The z-score for a value of a random variable is
  the number of standard deviations that x falls
  from the mean µ.
• It is calculated as

27
Example Finding Your Relative Standing on The SAT

• Scores on the verbal or math portion of the SAT
  are approximately normally distributed with mean
  µ 500 and standard deviation s 100. The
  scores range from 200 to 800.

28
Example Finding Your Relative Standing on The SAT

• If one of your SAT scores was x 650, how many
  standard deviations from the mean was it?

29
Example Finding Your Relative Standing on The SAT

• Find the z-score for x 650.

30
Example Finding Your Relative Standing on The SAT

• What percentage of SAT scores was higher than
  yours?
• Find the cumulative probability for the z-score
  of 1.50 from Table A.
• The cumulative probability is 0.9332.

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Example Finding Your Relative Standing on The SAT

• The cumulative probability below 650 is 0.9332.
• The probability above 650 is 1 0.9332
  0.0668
• About 6.7 of SAT scores are higher than yours.

32
Example What Proportion of Students Get A Grade
of B?

• On the midterm exam in introductory statistics,
  an instructor always give a grade of B to
  students who score between 80 and 90.
• One year, the scores on the exam have
  approximately a normal distribution with mean 83
  and standard deviation 5.
• About what proportion of students get a B?

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Example What Proportion of Students Get A Grade
of B?

• Calculate the z-score for 80 and for 90

34
Example What Proportion of Students Get A Grade
of B?

• Look up the cumulative probabilities in Table A.
• For z 1.40, cum. Prob. 0.9192
• For z -0.60, cum. Prob. 0.2743
• It follows that about 0.9192 0.2743 0.6449,
  or about 64 of the exam scores were in the B
  range.

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Using z-scores to Find Normal Probabilities

• If were given a value x and need to find a
  probability, convert x to a z-score using
• Use a table of normal probabilities to get a
  cumulative probability.
• Convert it to the probability of interest.

36
Using z-scores to Find Random Variable x Values

• If were given a probability and need to find the
  value of x, convert the probability to the
  related cumulative probability.
• Find the z-score using a normal table.
• Evaluate x zs µ.

37
Example How Can We Compare Test Scores That Use
Different Scales?

• When you applied to college, you scored 650 on an
  SAT exam, which had mean µ 500 and standard
  deviation s 100.
• Your friend took the comparable ACT in 2001,
  scoring 30. That year, the ACT had µ 21.0 and
  s 4.7.
• How can we tell who did better?

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What is the z-score for your SAT score of 650?

• For the SAT scores µ 500 and s 100.
• 2.15
• 1.50
• -1.75
• -1.25

39
What percentage of students scored higher than
you?

1. 10
2. 5
3. 2
4. 7

40
What is the z-score for your friends ACT score
of 30?

• The ACT scores had a mean of 21 and a standard
  deviation of 4.7.
• 1.84
• -1.56
• 1.91
• -2.24

41
What percentage of students scored higher than
your friend?

1. 3
2. 6
3. 10
4. 1

42
Standard Normal Distribution

• The standard normal distribution is the normal
  distribution with mean µ 0 and standard
  deviation s 1.
• It is the distribution of normal z-scores.