An Introduction to Business Statistics

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An Introduction to Business Statistics

• 1.1 Populations and Samples
• 1.2 Sampling a Population of Existing Units
• 1.4 Ratio, Interval, Ordinal, and Nominative
  Scales of Measurement

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Populations and Samples

• Population
• A set of existing units (people, objects, or
  events)
• Variable
• A measurable characteristic of the population
• Quantitative (real-valued)/ Qualitative
  (categorical)
• Sample
• A selected subset of the units of a population

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Populations and Samples

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Descriptive Statistics and Statistical Inference

• Descriptive Statistics
• The science of describing important aspects of a
  set of measurements
• Statistical Inference
• The science of using a sample of measurements to
  make generalizations about important aspects of a
  population

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Sampling a Population of Existing Units

• Random Sampling
• A procedure for selecting a subset of the
  population units in such a way that every unit in
  the population has an equal chance of selection
• Sampling with replacement
• When a unit is selected as part of the sample,
  its value is recorded and placed back into the
  population for possible reselection
• Sampling without replacement
• Units are not placed back into the population
  after selection

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Other Types of Samples
Frame A list of all population units. Required
for random sampling, but not for approximate
random sampling methods like systematic or for
voluntary response sampling. Systematic
Sample Every k-th element of the population is
selected for the sample Voluntary Response
Sample Sample units are self-selected (as in
radio/TV surveys)
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Scales of Measurement

• Ratio
• Quantitative scale, ratios are meaningful,
  inherently defined zero. (e.g. salary, height,
  distance.)
• Interval
• Quantitative scale, but ratios not meaningful
  nor is there an inherently defined zero. (e.g.
  temperature)
• Ordinal
• Qualitative or categorical scale with meaningful
  ordering or ranking of categories. (e.g. income
  classification)
• Nominative
• Qualitative scale without meaningful ordering
  among categories (e.g. gender, ethnic
  classification)

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Descriptive Statistics

• 2.1 Describing the Shape of a Distribution
• 2.2 Describing Central Tendency
• 2.3 Measures of Variation
• 2.4 Percentiles, Quartiles, and
  Box-and-Whiskers Displays
• 2.5 Describing Qualitative Data
• 2.6 Using Scatter Plots to Study the
  Relationship Between Variables

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Histograms
Example 2.4 The Accounts Receivable Case
Frequency Histogram
Relative Frequency Histogram
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The Normal Curve
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Skewness
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2.2 Population Parameters and Sample Statistics
A population parameter is number calculated from
all the population measurements that describes
some aspect of the population. The population
mean, denoted ?, is a population parameter and is
the average of the population measurements. A
point estimate is a one-number estimate of the
value of a population parameter. A sample
statistic is number calculated using sample
measurements that describes some aspect of the
sample.
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Measures of Central Tendency
Mean, m The average or expected value Median,
Md The middle point of the ordered
measurements Mode, Mo The most frequent value
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The Mean
Population X1, X2, , XN
Sample x1, x2, , xn
?
Population Mean
Sample Mean
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The Sample Mean
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Example Car Mileage Case
Example 2.5 Sample mean for first five car
mileages from Table 2.1 30.8, 31.7, 30.1,
31.6, 32.1
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The Median
The population or sample median is a value such
that 50 of all measurements lie above (or below)
it. The median Md is found as follows 1. If the
number of measurements is odd, the median is the
middlemost measurement in the ordered
values. 2. If the number of measurements is
even, the median is the average of the two
middlemost measurements in the ordered values.
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Example Sample Median
Example 2.6 Internists Salaries (x1000) 127
132 138 141 144 146 152 154 165 171 177 192
241 Since n 13 (odd,) then the median is the
middlemost or 7th measurement, Md152
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The Mode
The mode, Mo of a population or sample of
measurements is the measurement that occurs most
frequently.
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Relationships Among Mean, Median and Mode
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2.3 Measures of Variation
Range Largest minus the smallest
measurement Variance The average of the squared
deviations from the mean Standard Deviation The
square root of the variance
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The Range
Range largest measurement - smallest measurement
Example Internists Salaries (in thousands of
dollars) 127 132 138 141 144 146 152 154 165 171
177 192 241 Range 241 - 127 114 (114,000)
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The Variance
Population X1, X2, , XN
s2
Population Variance
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The Standard Deviation
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Example Population Variance/Standard Deviation
Population of annual returns for five junk bond
mutual funds 10.0, 9.4, 9.1, 8.3, 7.8
m 10.09.49.18.37.8 44.6 8.92
5 5
1.1664.2304.38441.2544 3.068
.6136 5
5
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Example Sample Variance/Standard Deviation
Example 2.11 Sample variance and standard
deviation for first five car mileages from Table
2.1 30.8, 31.7, 30.1, 31.6, 32.1
s2 2.572 ? 4 0.643