SAMPLING STATISTICS Since statistics are actually random
variables associated with a given sample, they will vary from
sample to sample. Therefore, they have probabilities distributions
associated with them. This will allows us to find probabilities
associated with the sample. i.e. what is the probability that the
mean of the population matches the mean of your sample.
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SAMPLING DISTRIBUTIONS ABOUT THE MEAN 1.Obtain a simple random
sample of size n. 2.Compute the sample mean. 3.Repeat steps 1 and 2
until all simple random samples have been obtained from the
population.
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EXAMPLE The weights of pennies minted after 1982 are
approximately normally distributed with mean 2.46 grams and
standard deviation 0.02 grams. Approximate the sampling
distribution of the sample mean by obtaining 200 simple random
samples of size n = 5 from this population.
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The data on the following slide represent the sample means for
the 200 simple random samples of size n = 5. For example, the first
sample of n = 5 had the following data: 2.493 2.466 2.473
2.4922.471 Note: = 2.479 for this sample
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EFFECT OF SAMPLE SIZE Repeat Experiment using sample size of n
= 20 The mean of the 200 sample means for n =20 is still 2.46, but
the standard deviation is now 0.0045 (0.0086 for n = 5). As
expected, there is less variability in the distribution of the
sample mean with n =20 than with n =5.
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EXAMPLE Suppose that the mean time for an oil change at a
10-minute oil change joint is 11.4 minutes with a standard
deviation of 3.2 minutes 1.If a random sample of n = 35 oil changes
I selected describe the sampling distribution of the sample mean.
2.If a random sample of n = 35 oil changes is selected, what is the
probability that the mean oil change time is less than 11
minutes.
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#21: THE LENGTH OF HUMAN PREGNANCIES IS APPROXIMATELY NORMALLY
DISTRIBUTED WITH MAN 266 DAYS AND STANDARD DEVIATION OF 16 DAYS
a)What is the probability a randomly selected pregnancy lasts less
than 260 days. b)Suppose a random sample of 20 pregnancies is
obtained. Describe the sampling distribution of the sample mean
length of human pregnancies. c)What is the probability that a
random sample of 20 pregnancies has a mean gestation period of 260
days or less? d)What is the probability that a random sample of 50
pregnancies has a mean gestation period of 260 days or less? e)What
might you conclude if a random sample of 50 pregnancies resulted in
a mean gestation period of 260 days or less?
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DISTRIBUTION OF SAMPLE PROPORTIONS Lesson 8.2
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POINT ESTIMATE OF A POPULATION PROPORTION
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EXAMPLE In a Quinnipiac University Poll conducted in May of
2008, 1,745 registered voters nationwide were asked whether they
approved of the way George W. Bush is handling the economy. 349
responded yes. Obtain a point estimate for the proportion of
registered voters who approve of the way George W. Bush is handling
the economy.
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8-17 According to a Time poll conducted in June of 2008, 42% of
registered voters believed that gay and lesbian couples should be
allowed to marry. Describe the sampling distribution of the sample
proportion for samples of size n=10, 50, 100. Using Simulation to
Describe the Distribution of the Sample Proportion
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8-21 Key Points from Example 2 Shape: As the size of the
sample, n, increases, the shape of the sampling distribution of the
sample proportion becomes approximately normal. Center: The mean of
the sampling distribution of the sample proportion equals the
population proportion, p. Spread: The standard deviation of the
sampling distribution of the sample proportion decreases as the
sample size, n, increases.
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SAMPLING DISTRIBUTION CHARACTERISTICS
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According to a Time poll conducted in June of 2008, 42% of
registered voters believed that gay and lesbian couples should be
allowed to marry. Suppose that we obtain a simple random sample of
50 voters and determine which believe that gay and lesbian couples
should be allowed to marry. Describe the sampling distribution of
the sample proportion for registered voters who believe that gay
and lesbian couples should be allowed to marry.
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COMPUTE PROBABILITIES OF SAMPLE PROPORTIONS According to the
Centers for Disease Control and Prevention, 18.8% of school-aged
children, aged 6-11 years, were overweight in 2004. (a)In a random
sample of 90 school-aged children, aged 6- 11 years, what is the
probability that at least 19% are overweight? (b)Suppose a random
sample of 90 school-aged children, aged 6-11 years, results in 24
overweight children. What might you conclude?
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#17: ACCORDING TO A USA TODAY SNAPSHOT, 26% OF ADULTS DO NOT
HAVE ANY CREDIT CARDS. A SIMPLE RANDOM SAMPLE OF 500 ADULTS IS
OBTAINED. 8-25