Download Ch. 10 Sampling Distributions 1. Parameter and statistic Recall

Ch. 10 Sampling Distributions
1. Parameter and statistic
Recall
• Population:
• Sample:
• Parameter: a number that describes
the population
• Statistic: a number that can be computed from the sample data without
making use of any unknown parameters
Example 10.1 We were interested in the
weight of 2001 Gala apples. We measured
the weights of 2000 2001 Gala apples. µ:
average weight of all 2001 Gala apples, x̄:
average weight of the 2000 2001 Gala apples. Which one is the parameter? Statistic?
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For Q,
• Parameter: µ, the population mean
• Statistic: x̄, the sample mean
In statistical practice, parameter is unknown;
we use a statistic to estimate an unknown
parameter.
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Example 10.2 State whether each number in the following is a parameter or a
statistic
(a) The Bureau of Labor Statistics last
month interviewed 60,000 members of
the labor force, of whom 7.2% were
unemployed.
(b) A telemarketing firm in Los Angeles
uses a device that dials residential telephone numbers in the city at random.
Of the first 100 numbers dialed, 48%
are unlisted. This is not surprising because 52% of all Los Angeles residential phones are unlisted.
(c) A researcher carries out a randomized
comparative experiment with young rats
to investigate the effects of a toxic
compound in food. She feeds the control group a normal diet. The experimental group receives a diet with 2500
parts per million of the toxic material.
After 8 weeks, the mean weight gain
is 335 grams for the control group and
289 grams for the experimental group.
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2. Sampling Distributions
Recall:
• population, sample
• parameter, statistic
For Q,
• Parameter: µ, the population mean;
and σ, the population s.d.
• Statistic: x̄, the sample mean; and s,
the sample s.d.
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Example 10.3 Fig. A shows the distribution of return for all 1815 stocks listed
on the New York Stock Exchange for the
entire year 1987. This was a year of extreme swings in stock prices, including a
record loss of over 20% in a single day.
The mean return for all 1815 stocks was
−3.5% and the distribution shows a very
wide spread. Fig. B shows the distribution
of returns for all possible portfolios that invested equal amounts in each of 5 stocks.
A portfolio is just a sample of 5 stocks
and its return is the average return for
the 5 stocks chosen. The mean return for
all portfolios is still −3.5%, but the variation among portfolios is much less than
the variation among individual stocks. For
example, 11% of all individual stocks had a
loss of more than 40%, but only 1% of the
portfolios had a loss that large. The two
Figures illustrate a basic principle of investment: diversification reduces risk. That
is, buying several securities rather than just
one reduces the variability of the return on
an investment.
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• Statistic varies from sample to sample
— sampling variability.
• Nevertheless, there is a regular distribution of its values in a large number
of repetitions.
Sampling distribution of a statistic: the
distribution of values taken by the statistic in all possible samples of the same size
from the same population.
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For SRS
• Mean of x̄ is equal to µ
• Averages are less variable than individual observations — as sample size
n increases, the s.d. of x̄ decreases,
√
in fact the s.d. of x̄ is σ/ n
• If the population distribution is normal, x̄ is always normal
• If the population distribution is not
normal, averages are more normal than
individual observations — when n is
large, the sampling distribution of x̄
is close to normal (the central limit
theorem). How large an n is needed
depends on the shape of the population distribution.
A statistic used to estimate a parameter is
unbiased if the mean of its sampling distribution is equal to the true value of the
parameter being estimated.
x̄ is an unbiased estimator of µ.
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x̄ − µ
√ ∼ N (0, 1)
σ/ n
if the population is normal;
x̄ − µ .
√ ∼ N (0, 1)
σ/ n
when n is large, no matter what shape the
population distribution has.
Thus, probability problems about x̄ are
the normal distribution calculation problems discussed in Chapter 3 with
z − score =
8
x̄ − µ
√
σ/ n
Example 10.4 The height x of young American women varies approximately according to N (64.5, 2.5).
P (x > 67) =
For an SRS of 10
P (x̄ > 67) =
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Example 10.5 The GPAs of all students
enrolled at a large university have an approximately normal distribution N (3.03, .28).
Find the probability that the mean GPA of
an SRS of 16 students selected from this
university is
(a) 3.1 or higher
(b) 2.89 or lower
(c) 2.89 to 3.1
Solution:
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Example 10.6 A life insurance company
sells a term insurance policy to a 21-year
old male that pays $100,000 if the insured
dies within the next 5 years. Suppose that
the distribution of profit for each policy has
mean $100 and standard deviation $9,000.
(a) If the insurance company sells 25 such
policies, what is the probability that
the average profit per policy is less
than -$1000?
(b) If the insurance company sells 4,000,000
such policies, what is the probability
that the average profit on the 4,000,000
policies is
(i) less than $90
(ii) more than $110
(iii) $90 to $110
Solution:
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Example 10.7 Children in kindergarten are
sometimes given the Ravin Progressive Matrices Test (RPMT) to assess their readiness for learning. Experience at Southwark Elementary School suggests that the
RPMT scores for its kindergarten pupils
have mean 13.6 and s.d. 3.1. The distribution is close to normal. Mr. Lavin has
22 children in his kindergarten class this
year. He suspects that their RPMT scores
will be unusually low because the test was
interrupted by a fire drill. To check this
suspicion, he wants to find the level L such
that there is probability only 0.05 that the
mean score of 22 children falls below L
when the usual Southwark distribution remains true. What is the value of L?
Solution:
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