MATH 166 Spring 2016
c
Wen
Liu
3.2
3.2 Measures of Central Tendency
Definition:
• The average or Mean of the n numbers x1 , x2 , . . . , xn , denoted by µ, is given by
µ=
x1 + x2 + · · · + xn
n
• The median of a set of numerical data is the middle number where the numbers are arranged
in order of size and there is an odd number of entries in the set. In the case that the number
of entries in the set is even, the median is the mean of the two middle numbers.
• The mode of a set of observations is the observation that occurs more frequently than the
others.
• Let X denote the random variable that has values x1 , x2 , . . . , xn , and let the associated probabilities be p1 , p2 , . . . , pn . The expected value or mean of the random variable X, denoted by
E(X), is
E(X) = x1 p1 + x2 p2 + · · · + xn pn
• The expected value of the binomial distribution with n trials and probability of success p in a
single trial and q = 1 − p is
E(X) = np
Examples:
1. You are thinking of buying a small manufacturer of quality yachts and have obtained the
following data from the owners that give how often various numbers of yacht sales occur per
quarter. Find the average or mean number sold per quarter.
Number Sold in One Quarter
Number of Quarters
1 2 3 4
10 15 16 7
5
2
2. Redo Example 1 using the formula for E(X).
Number Sold in One Quarter
Probability of Occurrence
1
2
3
4
5
10/50 15/50 16/50 7/50 2/50
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MATH 166 Spring 2016
3.2
c
Wen
Liu
Technology Corner: To compute the mean or median :
• If the data is listed as a set (for example, the set of data {1, 1, 1, 2, 3, 6, 6, 6}):
Press STAT , choose EDIT, and then enter the data values into list L1. Then press STAT ,
right arrow over to the CALC menu, choose 1:1-Var Stats, and press ENTER . The home page
should appear “1:1-Var Stats”. Press 2ND and 1 so that the screen shows “1:1-Var Stats L1 ”.
Press ENTER .
The mean is displayed as x.
The down arrow must be pressed to see the median, which is displayed as Med.
• If the data is in the form of a probability or frequency distribution:
Press STAT , choose EDIT, enter the X values into list L1, and then enter the probability or
frequency into list L2. Then press STAT , right arrow over to the CALC menu, choose 1:1-Var
Stats, and press ENTER . The home page should appear “1:1-Var Stats”. Press 2ND— 1 —a
comma—2ND— 2 so that the screen shows “1:1-Var Stats L1 , L2 ”. Press ENTER .
Examples:
1. Find the mean, median, and the mode of the set of data {1, 1, 1, 2, 3, 6, 6, 6}.
2. The frequency distribution of the hourly wage rates (in dollars) among blue-collar workers in a
certain factory is given in the following table. Find the mean (or average) wage rate, the mode,
and the median wage rate of these workers.
Wage Rate 10.40 10.50 10.60 10.70 10.80 10.90
Frequency
60
90
75
120
60
45
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MATH 166 Spring 2016
3.2
c
Wen
Liu
3. On the basis of past experience, the manager of the VideoRama Store has compiled the following
table, which gives the probabilities that a customer who enters the VideoRama Store will buy
0, 1, 2, 3, or 4 DVDs. How many DVDs can a customer entering this store be expected to buy?
DVDs
0
1
2
3
4
Probability 0.40 0.36 0.17 0.06 0.01
4. A man purchased a $20000, 1-year term-life insurance policy for $425. Assuming that the
probability that he will live for another year is 0.98, find the company’s expected gain.
5. A man wishes to purchase a life insurance policy that will pay the beneficiary $15000 in the event
that the man’s death occurs during the next year. Using life insurance tables, he determines
that the probability that he will live another year is 0.95. What is the minimum amount that he
can expect to pay for his premium? Hint: The minimum premium occurs when the insurance
company’s expected profit is zero.
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MATH 166 Spring 2016
3.2
c
Wen
Liu
6. A lottery has a grand prize of $180000, two runner-up prizes of $18000 each, twelve third-place
prizes of $9000 each, and twenty-four consolation prizes of $1800 each. If 720000 tickets are
sold for $1 each and the probability of any one ticket winning is the same as that of any other
ticket winning, find the expected return on a $1 ticket.
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