Sampling Distributions

1
Section 4.3

• Sampling Distributions

2
Population vs Sample

• Sample-the part of the population that is
  actually examined. Throughout the rest of the
  semester, we will assume that we have a random
  and unbiased sample.
• Statistic-a number that can be computed from the
  sample data. This number is used to estimate the
  parameter.

• Population-group we want information about.
• Parameter-a number that describes the population.
  Usually unknown.

3
Examples

• The mean of a population is _____. The statistic
  used to estimate _____ is __________________.
• The standard deviation of a population is _____.
  The statistic used to estimate _____ is
  ______________________________________.
• The proportion of successes in the population is
  _____. The statistics used to estimate _____ is
  _______________________________________.

4
Population My Pennies

• My pennies 2000 2000 2000 2000 1999
  1999 1998 1998 1998 1998 1998 1997 1996
  1996 1996 1995 1995 1995 1994 1993 1993
  1993 1992 1992 1991 1990 1990 1990 1990
  1988 1988 1987 1986 1986 1984 1984 1984
  1984 1984 1983 1983 1983 1982 1982 1981
  1980 1980 1979 1979 1976 1976 1975 1975
  1972 1969 1968 1968

5
The Law of Large Numbers

• Sample size 1 1982,
• Sample size 2 1982, 1999 ,
• Sample Size 3 1982, 1999, 1968,
• Law of Large Numbers -

6
Sample Size of Five

• My five pennies are dated ____, ____, ____,
  ____, ____. Sample mean_____.
• Sketch the histogram of pennies for class and the
  histogram of washers for class.

7
Sample Size of Five, Continued

• Histogram of pennies represents
• Describe shape, center, and spread of pennies
• We are trying to estimate
• The histogram of washers represents
• Describe shape, center, and spread of washers

8
Results for Sample Means

• Assume the population has mean m and standard
  deviation s. We use the sample mean of a
  randomly selected sample of size n to estimate m.
• The mean of the sampling distribution of the
  sample mean is __________.
• In other words the sample mean is an __________
  estimator of µ.
• The standard deviation of the sampling
  distribution of the sample mean is __________.
• Averages are less variable than individual
  observations.
• The results of large samples are less variable
  than the results of small samples.
• The CLT - The process of averaging smooths out
  the population distribution into an approximately
  normal distribution.
• If the population is itself normal, then the
  sampling distribution of the sample mean is
  normal for any sample size.
• More observations are required if the populations
  distribution is far from normal.

9
Example

• 1. The Wechsler Adult Intelligence Scale (WAIS)
  is a common IQ test for adults. The distribution
  of WIAS scores for persons over 16 years of age
  is approximately normal with mean 100 and
  standard deviation 15.
• What is the probability that a randomly chosen
  individual has a WAIS score of 105 or higher?
• What are the mean and standard deviation of the
  average WAIS score for a SRS of 60 people?
•  
• What is the probability that the average WAIS
  score of a SRS of 60 people is 105 or higher?
• Would your answers to any of the above parts be
  affected if the distribution of WAIS scores in
  the adult population were distinctly non-normal?