Sampling Distribution of the Sample Mean

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Sampling Distribution of the Sample Mean

• Behavior of the sample average

2
X-bar

• The situation in this section is that we are
  interested in the average of the population that
  has a certain characteristic.
• This average is the population parameter of
  interest, denoted by the greek letter mu.
• We estimate this parameter with the statistic
  x-bar, the average in the sample.

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X-bar Definition
4
Sampling Distribution of x-bar

• How does x-bar behave? To study the behavior,
  imagine taking many random samples of size n, and
  computing an x-bar for each of the samples.
• Then we plot this set of x-bars with a histogram.

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Sampling Distribution of x-bar
6
Central Limit Theorem

• The key to the behavior of x-bar is the central
  limit theorem. It says
• Suppose the population has mean, m, and standard
  deviation s. Then, if the sample size, n, is
  large enough, the distribution of the sample
  mean, x-bar will have a normal shape, the center
  will be the mean of the original population, m,
  and the standard deviation of the x-bars will be
  s divided by the square root of n.

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Central Limit Theorem

• If the CLT holds we have,
• Normal shape
• Center mu
• Spread sigma/sqroot n.

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When Does CLT Hold?

• Answer generally depends on the sample size, n,
  and the shape of the original distribution.
• General Rule the more skewed the population
  distribution of the data, the larger sample size
  is needed for the CLT to hold.

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CLT
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CLT and Sample Size

• Previous overhead shows the original population
  distribution in (a), and increasing sample sizes
  through graphs (b), (c), and (d).
• Notice that it takes large sample sizes (n30-35)
  for the distribution of x-bars to become normal
  for this very skewed population distribution.

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CLT and Sample Size

• If the population distribution is normal to start
  with, the distribution of x-bars will have a
  normal shape for all sample sizes.

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CLT
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Properties of x-bar

• When sample sizes are fairly large, the shape of
  the x-bar distribution will be normal by the CLT.
• The mean of the distribution is the value of the
  population parameter mu, m.
• The standard deviation of this distribution is
  the standard deviation of the population divided
  by square root of n.

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Computer Simulation

• Select chapter 5 computing activities to simulate
  the x-bar sampling distribution.
• Again note that the properties given earlier hold
  in the simulations.

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Calculate Probabilities

• Because the shape of the distribution is normal,
  we can standardize the variable x-bar to a Z
  standard normal distribution. Use Z-transform

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Example

• Cola bottles filled so that contents X have a
  normal distribution with mean298ml and standard
  deviation sigma3ml.
• What proportion of bottles have less than 295ml?
• Ans P(Xlt295)P(Zlt-1) .1586 by using the
  midterm 1 z-score formula.

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Example

• What is the probability of the average of a six
  pack of bottles being less than 295ml?
• Ans P(x-bar lt 295)

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Cola Example

• P(Xlt295) .15, but P(x-bar lt 295) .007
• As the sample size increases, the variation in
  the distribution decreases so that a value like
  295ml is very difficult and rare to occur in an
  average of a six pack or more of bottles, but
  could quite easily occur in a single bottle.
• Big point averages have less variation than
  individual observations.

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Diagram of Cola Problem
X-bar for Six-Pack
One Bottle
298
295
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Sampling Distribution of x-bar
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Random Rectangles
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Random Rectangles

• Select a representative sample of 5 rectangles,
  compute the sample average area x-bar for these
  rectangles.
• Next, select a simple random sample of 5
  rectangles. Select five two digit numbers from
  the random number table. Find their areas and
  then compute the average for your sample.
• Plot histograms of both averages representative
  and random sample.

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Random Rectangles

• Notice that the histogram of the x-bars (the
  sampling distribution of xbar) has an almost
  normal pattern (CLT of n5 works here), and
  notice the value at the center of the
  distribution. This should be close to the
  population mean.
• Which histogram has higher values? Why is this?
• Next overhead shows population average, sd, and
  shape of original distribution.

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Random Rectangles