4.4 Sampling Distribution Models and the Central Limit Theorem

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4.4 Sampling Distribution Models and the Central
Limit Theorem

• Transition from Data Analysis and Probability to
  Statistics

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Sampling Distributions

• Population parameter a numerical descriptive
  measure of a population.
• (for example ???? , p (a population proportion)
  the numerical value of a population parameter is
  usually not known)
• Example ? mean height of all NCSU students
• pproportion of Raleigh residents who favor
  stricter gun control laws
• Sample statistic a numerical descriptive measure
  calculated from sample data.
• (e.g, x, s, p (sample proportion))

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Parameters Statistics

• In real life parameters of populations are
  unknown and unknowable.
• For example, the mean height of US adult (18)
  men is unknown and unknowable
• Rather than investigating the whole population,
  we take a sample, calculate a statistic related
  to the parameter of interest, and make an
  inference.
• The sampling distribution of the statistic is the
  tool that tells us how close the value of the
  statistic is to the unknown value of the
  parameter.

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DEF Sampling Distribution

• The sampling distribution of a sample statistic
  calculated from a sample of n measurements is the
  probability distribution of values taken by the
  statistic in all possible samples of size n taken
  from the same population.
• Based on all possible samples of size n

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• In some cases the sampling distribution can be
  determined exactly.
• In other cases it must be approximated by using a
  computer to draw some of the possible samples of
  size n and drawing a histogram.

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Sampling distribution of p, the sample
proportion an example

• If a coin is fair the probability of a head on
  any toss of the coin is p 0.5.
• Imagine tossing this fair coin 5 times and
  calculating the proportion p of the 5 tosses that
  result in heads (note that p x/5, where x is
  the number of heads in 5 tosses).
• Objective determine the sampling distribution of
  p, the proportion of heads in 5 tosses of a fair
  coin.

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Sampling distribution of p (cont.) Step 1 The
possible values of p are 0/50, 1/5.2, 2/5.4,
3/5.6, 4/5.8, 5/51

• Binomial
• Probabilities
• p(x) for n5,
• p 0.5
• x p(x)
• 0 0.03125
• 1 0.15625
• 2 0.3125
• 3 0.3125
• 4 0.15625
• 5 0.03125

The above table is the probability distribution
of p, the proportion of heads in 5 tosses of a
fair coin.
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Sampling distribution of p (cont.)

• E(p) 0.03125 0.2.15625 0.4.3125 0.6.3125
  0.8.15625 1.03125 0.5 p (the prob of
  heads)
• Var(p)
• So SD(p) sqrt(.05) .2236
• NOTE THAT SD(p)

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Expected Value and Standard Deviation of the
Sampling Distribution of p

• E(p) p
• SD(p)
• where p is the success probability in the
  sampled population and n is the sample size

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Shape of Sampling Distribution of p

• The sampling distribution of p is approximately
  normal when the sample size n is large enough. n
  large enough means npgt10 and nqgt10

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Example

• 8 of American Caucasian male population is color
  blind.
• Use computer to simulate random samples of size n
  1000

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The sampling distribution model for a sample
proportion p Provided that the sampled values
are independent and the sample size n is large
enough, the sampling distribution of p is modeled
by a normal distribution with E(p) p and
standard deviation SD(p) , that
is where q 1 p and where n large enough
means npgt10 and nqgt10 The Central Limit
Theorem will be a formal statement of this fact.
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Example binge drinking by college students

• Study by Harvard School of Public Health 44 of
  college students binge drink.
• 244 college students surveyed 36 admitted to
  binge drinking in the past week
• Assume the value 0.44 given in the study is the
  proportion p of college students that binge
  drink that is 0.44 is the population proportion
  p
• Compute the probability that in a sample of 244
  students, 36 or less have engaged in binge
  drinking.

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Example binge drinking by college students
(cont.)

• Let p be the proportion in a sample of 244 that
  engage in binge drinking.
• We want to compute
• E(p) p .44 SD(p)
• Since np 244.44 107.36 and nq 244.56
  136.64 are both greater than 10, we can model the
  sampling distribution of p with a normal
  distribution, so

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Example binge drinking by college students
(cont.)
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Example texting by college students

• 2008 study 85 of college students with cell
  phones use text messageing.
• 1136 college students surveyed 84 reported that
  they text on their cell phone.
• Assume the value 0.85 given in the study is the
  proportion p of college students that binge
  drink that is 0.85 is the population proportion
  p
• Compute the probability that in a sample of 1136
  students, 84 or less use text messageing.

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Example texting by college students (cont.)

• Let p be the proportion in a sample of 1136 that
  text message on their cell phones.
• We want to compute
• E(p) p .85 SD(p)
• Since np 1136.85 965.6 and nq 1136.15
  170.4 are both greater than 10, we can model the
  sampling distribution of p with a normal
  distribution, so

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Example texting by college students (cont.)
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Another Population Parameter of Frequent
Interest the Population Mean µ

• To estimate the unknown value of µ, the sample
  mean x is often used.
• We need to examine the Sampling Distribution of
  the Sample Mean x
• (the probability distribution of all possible
  values of x based on a sample of size n).

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Example

• Professor Stickler has a large statistics class
  of over 300 students. He asked them the ages of
  their cars and obtained the following probability
  distribution
• x 2 3 4 5 6 7 8
• p(x) 1/14 1/14 2/14 2/14 2/14 3/14 3/14
• SRS n2 is to be drawn from pop.
• Find the sampling distribution of the sample mean
  x for samples of size n 2.

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Solution

• Total of 7249 possible samples of size 2
• All 49 possible samples with the corresponding
  sample mean are on p. 35

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Solution (cont.)

• Probability distribution of x
• x 2 2.5 3 3.5 4
  4.5 5 5.5 6 6.5 7
  7.5 8
• p(x) 1/196 2/196 5/196 8/196 12/196
  18/196 24/196 26/196 28/196 24/196 21/196
  18/196 1/196
• This is the sampling distribution of x because it
  specifies the probability associated with each
  possible value of x
• From the sampling distribution above
• P(4 ? x ? 6) p(4)p(4.5)p(5)p(5.5)p(6)
• 12/196 18/196 24/196 26/196
  28/196 108/196

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Expected Value and Standard Deviation of the
Sampling Distribution of x
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Example (cont.)

• Population probability dist.
• x 2 3 4 5 6 7 8
• p(x) 1/14 1/14 2/14 2/14 2/14 3/14 3/14
• Sampling dist. of x
• x 2 2.5 3 3.5 4 4.5
  5 5.5 6 6.5 7 7.5 8
• p(x) 1/196 2/196 5/196 8/196 12/196 18/196
  24/196 26/196 28/196 24/196 21/196 18/196
  1/196

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• Population probability dist.
• x 2 3 4 5 6 7 8
• p(x) 1/14 1/14 2/14 2/14 2/14 3/14 3/14
• Sampling dist. of x
• x 2 2.5 3 3.5 4 4.5
  5 5.5 6 6.5 7 7.5 8
• p(x) 1/196 2/196 5/196 8/196 12/196
  18/196 24/196 26/196 28/196 24/196 21/196
  18/196 1/196

E(X)2(1/14)3(1/14)4(2/14) 8(3/14)5.714
Population mean E(X)? 5.714
E(X)2(1/196)2.5(2/196)3(5/196)3.5(8/196)4(12/
196)4.5(18/196)5(24/196) 5.5(26/196)6(28/196)
6.5(24/196)7(21/196)7.5(18/196)8(1/196) 5.714
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Example (cont.)
SD(X)SD(X)/?2 ?/?2
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IMPORTANT
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Sampling Distribution of the Sample Mean X
Example

• An example
• A die is thrown infinitely many times. Let X
  represent the number of spots showing on any
  throw.
• The probability distribution
• of X is

E(X) 1(1/6) 2(1/6) 3(1/6) 3.5 V(X)
(1-3.5)2(1/6) (2-3.5)2(1/6) . 2.92

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• Suppose we want to estimate m from the mean of
  a sample of size n 2.
• What is the sampling distribution of in this
  situation?

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6/36 5/36 4/36 3/36 2/36 1/36
1 1.5 2.0 2.5 3.0 3.5
4.0 4.5 5.0 5.5 6.0
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1
6
1
6
1
6
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The variance of the sample mean is smaller
than the variance of the population.
Mean 1.5
Mean 2.5
Mean 2.
1.5
2.5
Population
2
1
2
3
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
Compare the variability of the population to the
variability of the sample mean.
1.5
2.5
Let us take samples of two observations
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
1.5
2.5
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
Also, Expected value of the population (1 2
3)/3 2
Expected value of the sample mean (1.5 2
2.5)/3 2
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Properties of the Sampling Distribution of x
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Unbiased
Unbiased
Confidence
l
Precision
l
The central tendency is down the center
BUS 350 - Topic 6.1
6.1 -
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Handout 6.1, Page 1
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Consequences
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We Know More!

• We know 2 parameters of the sampling distribution
  of x