Math 145

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Math 145

• June 19, 2007

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Outline

• Recap
• Sampling Designs
• Graphical methods

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Statistics

• is the science of collecting, analyzing,
  interpreting, and presenting data.
• Two kinds of Statistics
• Descriptive Statistics.
• Inferential Statistics.
• Population
• Sample
• ? representative sample

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Methods of Acquiring Information

• Census
• Sampling
• Experimentation
• Observational Study researchers observe
  characteristics and take measurements, as in
  sample survey. (Association)
• Designed Experiment researchers impose
  treatments and controls and then observe
  characteristics and take measurements. (Cause and
  Effect)
• Consider 1.27 (p.22), 1.29

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Sampling Designs

• Simple Random Sampling.
• Systematic Random Sampling.
• Cluster Sampling.
• Stratified Random Sampling with Proportional
  Allocation.

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Simple Random Sampling

• A sampling procedure for which each possible
  sample of a given size has the same chance of
  being selected.
• Population of 5 objects A, B, C, D, E
• Take a sample of size 2.
• Possible samples (A,B), (A,C), (A,D), (A,E),
  (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)
• Random number generators

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Systematic Random Sampling

• Step 1. Divide the population size by the sample
  size and round the result down to the nearest
  number, m.
• Step 2. Use a random-number generator to obtain a
  number k, between 1 and m.
• Step 3. Select for the sample those numbers of
  the population that are numbered k, km, k2m,
• Expected number of customers 1000
• Sample size of 30 ? m 1000/30 33.33 ? 33
• Suppose k 5. Then select 5, 533, 566,

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Cluster Sampling

• Step 1. Divide the population into groups
  (clusters).
• Step 2. Obtain a simple random sample of
  clusters.
• Step 3. Use all the members of the clusters in
  step 2 as the sample.

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Stratified Random Sampling with Proportional
Allocation

• Step 1. Divide the population into subpopulations
  (strata).
• Step 2. From each stratum, obtain a simple random
  sample of size proportional to the size of the
  stratum.
• Step 3. Use all the members obtained in Step 2 as
  the sample.
• Population of 9,000 with 60 females and 40
  males
• Sample of size 80.
• ? 48 females (from 5,400) and 32 males (from
  3,600).

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Descriptive Statistics

• Individuals are the objects described by a set
  of data. Individuals may be people, but they may
  also be animals or things.
• Variable a characteristic of an individual. A
  variable can take different values for different
  individuals.
• Categorical (Qualitative) variable places an
  individual into one of several groups or
  categories. Gender, Blood Type
• Quantitative variable takes numerical values
  for which arithmetic operations such as adding
  and averaging make sense. Height, Income, Time,
  etc.
• Consider 1.18 (p. 20), 1.21 (p.21)

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Quantitative Variables

• Discrete Variables There is a gap between
  possible values.
• Counts (no. of days, no. of people, etc.)
• Age in years
• Continuous Variables Variables that can take on
  values in an interval.
• Survival time, amount of rain in a month,
  distance, etc.

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Graphical Procedures

• Categorical (Qualitative) Data
• Bar Chart
• Pie Chart
• Quantitative Data
• Histogram
• Stem-and-leaf plot (Stemplot)
• Dotplot
• These plots describe the Distribution of a
  variable.

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Length of Stay
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Fifth-grade IQ Scores
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Distribution

• - The distribution of a variable tells us what
  values it takes and how often it takes these
  values
• Categorical Data
• Table or Bar Chart
• Quantitative Data
• Frequency Table
• Histogram
• Stem-and-leaf plot

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Describing a distribution

• Skewness
• Symmetric
• Skewed to the right (positively skewed)
• Skewed to the left (negatively skewed)
• Center/Spread
• No of peaks (modes)
• Unimodal, Bimodal, Multimodal.
• Outliers
• Extreme values.

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Homework
Exercises Chapter 1 (pp. 19-23) 1, 2, 5,
11, 12, 16, 24, 28 Chapter 2 (pp. 36-40) 5,
6, 10. (pp. 50-53) 25, 30, 32.
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Thank you!