Exercise 19: Sample Size

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Exercise 19 Sample Size
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Part One

• Explore how sample size affects the distribution
  of sample proportions
• This was achieved by first taking random samples
  20 times when n10 and then taking 20 random
  samples where n40. These random samples were
  then summarized as sample statistics (p-hat).

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Tally for Discrete Variable Live

• Live Count Percent
• off 223 50.11
• on 222 49.89
• N 445
• 1
• This verifies that the proportion of students
  living on campus and off campus is approximately
  50. This would be the population proportion (p).

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Mean, Shape Standard Deviation

• What would you expect if 20 random samples of 10
  were taken?
• What would you expect if 20 random samples of 40
  were taken?

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Results from 20 samples where n10 resulting in
phatlive
0.6000 0.5000 0.5000 0.4000 0.5000 0.5556 0.7000 0
.4000 0.6000 0.8000

• 0.3000
• 0.4000
• 0.5000
• 0.4000
• 0.5000
• 0.4000
• 0.5000
• 0.3000
• 0.5000
• 0.6000

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Descriptive Statistics phatlive10

• Variable N N Mean SE Mean
  StDev
• Phatlive 20 0 0.4978
  0.0278 0.1242
• Minimum Q1 Median Q3
  Maximum
• 0.3000 0.4000 0.5000 0.5889
  0.8000

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Lets Look At A Stem Plot

• Stem-and-leaf of phatlive10 (N 20)
• Leaf Unit 0.010
•  
•   3 00
• 3
• 4 00000
• 4
• 5 0000000
• 5 5
• 6 000
• 6
• 7 0
• 7
• 8 0

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Sample Proportions

• What is the center, spread and shape for this
  sample proportion?
• Center mean 0.4978 phat
• Spread st.dev 0.1242
• Shape np and/or n(1-p) does not equal atleast
  10, therefore guidelines for normality are not
  met. However, as shown in the stem plot, the
  results appear relatively normal because of the
  perfectly balanced population proportions of .5
  and .5.

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What if the sample size increases

• Results from 20 samples where n40 resulting in
  phatlive

0.5750 0.4750 0.4500 0.4250 0.4750 0.3250 0.4250 0
.4000 0.4250 0.3500
0.5500 0.5000 0.5385 0.4359 0.4500 0.5000 0.4750 0
.4250 0.4500 0.4750
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Descriptive Statistics phatlive40

• Variable N N Mean SE Mean
  StDev
• Phatlive40 20 0 0.4562 0.0137
  0.0611
• Minimum Q1 Median Q3
  Maximum
• 0.3250 0.4250 0.4500 0.4938
  0.5750

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Stem-plot for phatlive40

• N 20 Leaf Unit 0.010
• 3 2
• 3 5
• 3
• 3
• 4 0
• 4 22223
• 4 555
• 4 7777
• 4
• 5 00
• 5 3
• 5 5
• 5 7

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Sample Proportions for phatlive40

• What is the center, spread and shape for this
  sample proportion?
• Center mean.4562
• Spread st. dev. .0611
• Shape np and n(1-p) are greater then 10 there
  normality satisfied.

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Lets compare them simultaneously

•  
• Descriptive Statistics phatlive40, phatlive10
•  
• Variable N N Mean SE Mean
  StDev Minimum Q1 Median
• phatlive40 20 0 0.4562 0.0137
  0.0611 0.3250 0.4250 0.4500
• phatlive10 20 0 0.4978 0.0278
  0.1242 0.3000 0.4000 0.5000
• Variable Q3 Maximum
• phatlive40 0.4938 0.5750
• phatlive10 0.5889 0.8000
•  How do their centers, spreads and shapes
  compare?

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Box-plots
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What does this mean?

• The mean for n40 is more consistent with the
  population mean.
• The spread is smaller for n40
• The shape is more normal for n40

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As outlined in Chapter 6

• A random variable X for count of sampled
  individuals in the category of interest is
  binomial with parameters n and p if
• There is a fixed sample size n
• Each selection is independent of the others
• Each individual sampled takes just two possible
  values
• The Probability of each individual falling in the
  category of interest is always p.

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However

• The second condition isnt really met when
  sampling without replacement. But as long as the
  population is at least 10n, then approximate
  independence can still be concluded.
• Since the population is greater then 400, both
  sample sizes of 10 and 40 follow this rule.

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Part 2

• Explores how population shape affects the
  distribution of sample proportion.
• First, 20 random samples of 10 were taken and
  then 20 random samples of 40 were taken. The
  results were compared.

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Handedness

• Tally for Discrete Variables Handed
•  
• Handed Count Percent
• ambid 13 2.91
• left 40 8.97
• right 393 88.12
• N 446
• Proportion of ambidextrous is very skewed since
  only approximately 3 of population is vs. 97
  who is not.

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For Handedness n10

• Variable N N Mean
  SE Mean
• phathandedn10 20 0 0.0300 0.0164
• StDev Min. Q1 Median Q3
  Max.
• 0.0733 0.00 0.00 0.00 0.00
  0.3000

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Stem-plot n10

• Stem-and-leaf of phathandedn10
• N 20 Leaf Unit 0.010
•  
• 0 0000000000000000
• 1 000
• 2
• 3 0

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What does this data show?

• The center or mean is 0.0300
• The spread is .0073
• The shape is not normal because the guidelines of
  np and n(1-p) being greater then 10 are not met

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Handedness n40

• Descriptive Statistics phathandedn40
•  
• Variable N N Mean SE
  Mean StDev
• phathandedn40 20 0 0.04000 0.00612
  0.02739
• Minimum Q1 Median Q3 Maximum
• 0.00000 0.02500 0.03750 0.05000 0.10000
•  

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Stem-plot n-40

•  
• Stem-and-leaf of phathandedn40 N 20
• Leaf Unit 0.0010
•  
• 0 000
• 1
• 2 5555555
• 3
• 4
• 5 000000
• 6
• 7 555
• 8
• 9
• 10 0

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What does this mean?

• The center or mean is 0.0400
• The spread is 0.02739
• The shape is normal because the guidelines of np
  and n(1-p) being greater then 10 are met.

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Lets compare them

• Variable N N Mean
  SE Mean StDev
• phathandedn40 20 0 0.0400 0.00612
  0.02739
• phathandedn10 20 0 0.0300 0.0164
  0.0733
• Minimum Q1 Median Q3
  Maximum
• 0.00000 0.02500 0.03750 0.05000
  0.10000
• 0.0000 0.0000 0.0000 0.0000
  0.3000
•  

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Lets compare them
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What does it mean?

• By increasing the sample size, the box plot
  became less skewed.
• There was less of a spread and fewer outliers.
• The center remained at approximately .03
• The shape became more normal.

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Overall

• Live seemed to be more normal the handedness.
  This was because the population was no skewed for
  the live variable like for handedness.
• In both situation, n40 caused the distributions
  to be more normal.