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Sampling Distribution of the Sample Mean

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What is the probability of the average of a six pack of bottles being less than 295ml? ... and rare to occur in an average of a six pack or more of bottles, but could ... – PowerPoint PPT presentation

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Title: Sampling Distribution of the Sample Mean


1
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.

3
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.

5
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.

7
Central Limit Theorem
  • If the CLT holds we have,
  • Normal shape
  • Center mu
  • Spread sigma/sqroot n.

8
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.

9
CLT
10
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.

11
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.

12
CLT
13
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.

14
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.

15
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

16
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.

17
Example
  • What is the probability of the average of a six
    pack of bottles being less than 295ml?
  • Ans P(x-bar lt 295)

18
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.

19
Diagram of Cola Problem
X-bar for Six-Pack
One Bottle
298
295
20
Sampling Distribution of x-bar
21
Random Rectangles
22
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.

23
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.

24
Random Rectangles
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