Sampling Distribution Models for MEANS

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Chapter 18

• Sampling Distribution Models for MEANS

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The sampling distribution

• Weve studied already the sampling distribution
  for proportions, and learned that it was
• Normal, with a mean (mu) at
• p, the true population proportion
• and
• a standard deviation of

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Sampling distributions

• Recall that a sampling distribution is
• The distribution of values taken on by a
  statistic (phat or xbar) in ALL possible samples
  of the same size (n) from the same population.
• The sampling distribution for MEANS works very
  similarly to what we saw for proportions.
• Lets look at the Penny Sample file and see what
  happens when we approximate a sampling
  distribution with different sample sizes.

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Pennies
The POPULATION distribution, mu1993.37,
sigma9.66972
Histogram of 1000 samples of size 7. mu
of distribution 1993.52 sigma
3.5002 Note shows a fair amount of skewness to
the left still
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More Pennies
Histogram of 1000 samples of size 21. mu
of distribution 1993.42 sigma
2.09872 Still slightly skewed left.
Histogram of 1000 samples of size 71. mu
of distribution 1993.42 sigma 1.16528
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The Central Limit Theorem
What did you notice about the histograms of the
sample means, as the sample size (n) got
larger? 1)They started looking closer and
closer to a Normal distribution!! 2) The mean of
the distribution was always near the true
population mean (mu) 3) The standard deviation
of the distributions got smaller as n
increased. These observations are what the
Central Limit Theorem tells us No matter what
the distribution of the underlying population is
(normal, skewed, multimodal, etc), the sampling
distribution of xbar gets closer and closer to
Normal as n increases.
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More on the Central Limit Theorem
The CLT (Central Limit Theorem) tells us that the
Normal model that the sampling distribution
approaches has 1)Mean (m(xbar)) EQUAL TO the
population mean, mu. 2) Standard Deviation
Did we see this with the Pennies? 1) mean of
histogram for n71 was 1993.42, extremely close
to the population mu of 1993.37. 2) SD of
histogram for n71 was 1.16528. By the CLT, it
should have been , so again, we were very
close. (remember, the histograms for the pennies
are not exactly sampling distributions, just
histograms with a LOT of samples)
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So what does this mean?
The center of the sampling distribution for xbar
will always be the same as the population mean.
The SD of xbar gets smaller as sample size
increases. (at a rate of sqrt(n)) The shape of
the sampling distribution gets more and more
Normal as n increases.
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Assumptions

• Of course, as always, there are assumptions that
  must be made to use the Normal model for sample
  means.
• Data must be a random sample
• Sampled values must be independent.
• Sample size lt 10 of population.
• How large of a sample is needed?
• This depends on the population.
• In general, if the population is Normal, we
  dont need to worry about the size of the sample.

If the population is NOT Normal, though, we do.
Rule of Thumb If ngt40, we can assume a Normal
Model. If 15ltnlt40, use Normal if
histogram of the sample shows little skewness, no
extreme outliers If nlt15, use Normal if
histogram of the sample is symmetric and no
outliers.
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Example Problem, 26
Ithaca, NY gets an average of 35.4 of rain each
year, with a standard deviation of 4.2. Assume
that a Normal model applies (to the population of
years) During what percentage of years does
Ithaca get more than 40 of rain? ie,
P(xgt40)? Note this part is just a
Chapter 6 question!!
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26 continued
Let xbar be the mean amount of rain for a random
sample of 4 years. Describe the sampling
distribution model of xbar. (check
assumptions!) What is the probability of a
sample of 4 years having less than 30 of
rain? ie, find P(xbarlt30).
Calculate z-score Look z up on table
P(xbarlt30)
What is the probability of a sample of 4 yrs
having more than 40? P(xbargt40)?
Z-score P(xbargt40)
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28 At work

• Some business analysts estimate that the length
  of time people work at a job has mean 6.2 yrs, SD
  4.5 yrs.
• Do you suspect this distribution is normal,
  skewed left, or skewed right?
• Why could you estimate the probability that 100
  people selected at random had worked an average
  10 yrs, but not do so for an individual (or even
  a sample of 3 or 4)

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36 Potato Chips

• The weight of potato chips in a medium size bag
  is stated to be 10 ounces. The amount that the
  packaging machine puts in these bags is believed
  to have a Normal model with mean 10.2 ounces, SD
  0.12 ounces.
• Some chips are sold in bargain packs of 3 bags.
  What is the probability that the mean weight of
  the 3 bags is below the stated amount?