Chapter 7 Sampling Distribution

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Chapter 7 Sampling Distribution

• I. Basic Definitions
• II. Sampling Distribution of Sample Mean
• III. Sampling Distribution of Sample Proportion
• IV. Odds and Ends
• I. Basic Definitions
• Population and sample (N presidents and n
  presidents)
• Parameter and statistic p.258p.264
• Based on all (N) possible values of X ?, ?
  and p
• Based on n values of X
• Statistic as a point estimator for parameter
• Sample statistic becomes a new random variable
• Sampling distribution probability distribution
  of a statistic

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II. Sampling Distribution of Sample Mean 1.
Summary measures p.270
- If N is infinite or N
gtgt n p.271 - n?? ? If X is
normally distributed, is normally
distributed. 2. Central Limit Theorem p.272 For
any distribution of population, if sample size is
large, the sampling distribution of sample mean
is approximately a normal distribution. 3.
Applications
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Example. P.278 18 A population has a mean of
200 and a standard deviation of 50. A simple
random sample of size 100 will be taken and the
sample mean will be used to estimate the
population mean. a. What is the expected value of
? b. What is the standard deviation of
? c. Show the sampling distribution of ? d.
What does the sampling distribution of
show? Answer. a. 200 b. c.
? N(200, 5) d. Probability distribution of

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3. Applications (1) Procedure ? Z ? P(Z)
? P( ) P(Z) Example. A population
has a mean of 100 and a standard deviation of 16.
What is the probability that a sample mean will
be within ?2 of the population mean for each of
the following sample sizes? a. n 50 b. n
200 c. What is the advantage of larger sample
size?
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Solution. Given ?x 100, ?x 16 a. n 50,
find From Z-Table When z -.88, the left
tail is .1894. b. n 200, find z2 1.77.
Use Z-Table
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c. Larger sample size results in a smaller
standard error - a higher probability that the
sample mean will be within ?2 of ?x
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(2) Procedure ? P(Z) ? Z ? Example.
The diameter of ping-pong balls is approximately
normally distributed with a mean of 1.30 inches
and a standard deviation of 0.04 inches. a. What
is the probability that a randomly selected
ping-pong ball will have a diameter between
1.28 and 1.30 inches? b. If a random sample of
16 balls are selected, 60 of the sample means
would be between what two values that are
symmetrically around population mean.
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Answer Random variable X diameter of a
ping-pong ball. a. P(1.28 ? X ? 1.3) .1915
z (1.28-1.3)/.04 .5 b. From Z table When
the left tail is .2005, z1 -.84 z2 .84
.5-.3085.1915
.2
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Homework Sample Mean p.278 18,
19 p.278 25
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III. Sampling Distribution of Sample Proportion
1. Summary measures

p.280 -
If N is infinite or N gtgt n p.280 -
n?? ? If n and p satisfy the rule of five,
is approximately normally distributed. P.282
2. Applications
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2. Applications (1) Procedure ? Z ? P(Z)
? P( ) P(Z) Example. Assume that
15 of the items produced in an assembly line
operation are defective, but that the firms
production manager is not aware of this
situation. Assume further that 50 parts are
tested by the quality assurance department to
determine the quality of the assembly operation.
Let be the sample proportion found defective
by the quality assurance test.
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a. Show the sampling distribution for . b.
What is the probability that the sample
proportion will be within ?.03 of the
population proportion that is defective? c. If
the test show .10 or more, the assembly
line operation will be shut down to check for
the cause of the defects. What is the
probability that the sample of 50 parts will
lead to the conclusion that the assembly line
should be shut down? Answer. a.
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b. c.
.2776
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Homework Sample Proportion p.283 31
p.284 35 p.285 40 Reading Compare wordings
in Chapter 6 homework problems, homework problems
for sample mean and homework problems for sample
proportion X?
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IV. Odds and Ends 1. Three properties of a good
estimator p.286p.287 (1) Unbiasedness Mean
of a sample statistic is equal to the population
parameter. (
) (2) Efficiency The estimator with smaller
standard deviation. (
) (3) Consistency If (The
values of sample statistic tend to become closer
to the population parameter. For example,

)
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2. Sampling Methods (p.260) and fpc (finite
population correction factor)
p.271p.280 (1) Infinite population, or finite
population sampling with replacement, or n/N ?
0.05 (2) Finite population sampling without
replacement and n/N gt 0.05
(fpc ? 1. If N gtgt n, fpc ? 1)