Introduction to Basic Statistics

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Topic 6
Marketing Analysis Research (MAR3613) By
Kanghyun Yoon

• Introduction to Basic Statistics

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Importance of Sampling (I)

• Why is the sampling procedure important?
• To make conclusions about the whole population
  (e.g., estimating some unknown characteristic of
  the population) by using a subset (e.g., sample)
  of the population.
• Example Use point or interval estimates (e.g.,
  confidence intervals) to estimate population
  parameters.
• Population vs. Sample
• Population or universe The entire group that
  shares some common set of characteristics (e.g.,
  people, sales territories, stores, students,
  etc.).
• It is important to prepare a sampling frame.
• Sample A subset of all the elements in the
  population.
• Census An investigation of all the individual
  elements in the population.
• It is useful for industrial product.
• Sample statistics vs. population parameters for
  defining the characteristics.
• Population parameters N, m, s, p (or p)
• Sample statistics n, x-bar, s, p-hat

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Importance of Sampling (II)

• Two Purposes of Sampling
• Descriptive statistics Describe the population
  in terms of key characteristics.
• Use frequency distribution, measures of central
  tendency, and measures of dispersion (or spread).
• Inferential statistics Make inferences about the
  characteristics of population.
• Implement hypothesis testing procedure.
• Important Terminologies
• Frequency distribution A summary table that
  describes the number of times that a particular
  value of a variable occurs (e.g., histogram).
• Example One-way frequency table, two-way
  crosstabulation, and so on.
• Measures of central tendency It indicates the
  center of the frequency distribution.
• Example The mean, median (e.g., the mid-point),
  and mode.
• Measures of dispersion It indicates how far each
  observation departs from the center of the
  frequency distribution.
• Example The range, variance, and standard
  deviation.

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Importance of Sampling (III)

• Three Types of Distributions
• Population distribution A frequency distribution
  constructed by using all the elements in the
  population.
• Sample distribution A frequency distribution
  constructed by using all the elements in the
  sample.
• Sampling distribution of the sample mean A
  theoretical probability distribution that uses
  sample means for all possible samples of a
  certain size drawn from a particular population.
• Central Limit Theorem
• As sample size increases, the distribution of
  sample means of size n approaches a normal
  distribution with a mean equal to ? and a
  standard deviation equal to s/vn.

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Relationship Among Distributions
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Normal Distribution

• Normal Distribution
• It is symmetrical and bell-shaped around its
  mean.
• The highest point indicates the mean, the median,
  and the mode.
• The area under the curve has a probability
  density to equal to one.
• Standardized Normal Distribution (SND)
• Key features it has a mean of zero and a
  standard deviation of one.
• About 68.26 of the observations will fall within
  one standard deviation of the mean.
• About 95.44 of the observations will fall within
  two standard deviation of the mean.
• Almost all of the observations (e.g., 99.9) will
  fall within three standard deviation of the mean.
• How to Transform Normal Distribution into SND?
• How to get the standardized normal distribution
  (e.g., z distribution)?
• Formula Z-score (Xi µ) / s.

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Linear Transformation
s
s
m
X
m
Sometimes the scale is stretched
Sometimes the scale is shrunk
-2 -1 0 1 2
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Basics to Hypothesis Testing

• Regarding Inferential Statistics
• Estimate the shape of population distribution
  using the sample information.
• Compare two frequency distributions of both the
  sample and the population.
• Estimate the central tendency of the population
  using the sample information.
• Utilize the point estimates or the confidence
  interval as of the interval estimates.
• Estimate the degree of dispersion of the
  population using the sample information.
• Compare both dispersion measures of the
  population and the sample.
• Hypothesis testing procedure is necessary.
• Point estimates vs. Interval estimates
• Point estimate It is an estimate of the
  population mean in the form of the sample mean.
• Interval estimate We use the confidence interval
  by using the following formula
• µ X-bar Random Error x-bar Z-score
  standard error of the mean (s/vn).

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Basics to Hypothesis Testing

• Why hypothesis testing procedure is required?
• Hypothesis is an assumption about the
  characteristics of the population being
  investigated.
• The researcher should determine whether a
  hypothesis regarding the characteristics of the
  population is likely to be true, by using the
  sample information, given an objective decision
  rule.
• An objective decision rule is related to the
  concept of statistical difference.
• Mathematical difference vs. statistical
  significance
• Consider the following equation Sales a0 b1
  Advertising
• H0 b1 0
• H1 b1 ? 0
• Is the observed difference (e.g., between b10.0
  in the population and b12.0 in the sample)
  likely to occur due to sampling error (e.g., by
  chance) or occur since b1 has 2.0 actually in the
  population?
• Key statistical concepts for an objective
  decision rule
• Null hypothesis (H0) vs. alternative hypothesis
  (H1 or Ha)
• Sample test statistics vs. critical value, given
  the choice of sampling distribution
• P-value vs. significance level of a ( critical
  region)
• Confidence level The probability that a
  particular interval (e.g., confidence interval as
  an interval estimate) will include the true
  population value.

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Basics to Hypothesis Testing

• Types of Sampling Distributions and Test
  Statistics
• Normal distribution when Z test statistic lies
  between negative and positive infinity (-? ? Z ?
  ?).
• Use Z test statistics in comparison of one-sample
  distribution or two-sample distributions.
• t distribution when t test statistic lies
  between negative and positive infinity (-? ? t ?
  ?).
• Use t test statistics in t-test analysis or
  regression analysis.
• F distribution when F test statistic (e.g.,
  the proportion) lies between zero and positive
  infinity (0 ? F ? ?).
• Use F test statistics in ANOVA analysis or
  regression analysis.
• Chi-Square (?2) distribution when chi-square
  test statistic (e.g., the square) lies between
  zero and positive infinity (0 ? ?2 ? ?).
• Use ?2 test statistics in frequency analysis or
  cross-tabulation analysis for chi-square test.