A Review of Basic Concepts

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A Review of Basic Concepts

• Chapter 1

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Definition 1.1

• Statistics is the science of data. This involves
  collecting, classifying, summarizing, organizing,
  analyzing, and interpreting data.

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Definition 1.2

• An experimental unit is an object (person or
  thing) upon which we collect data.

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Definition 1.3

• A variable is a characteristic (property) of the
  experimental unit with outcomes (data) that vary
  from one observation to the next.

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Definition 1.4

• Quantitative data are observations measured on a
  naturally occurring numerical scale.

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Definition 1.5

• Nonnumerical data that can only be classified
  into one of a group if categories are said to be
  qualitative data.

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Definition 1.6

• A population data set is a collection (or set) of
  data measured on all experimental units of
  interest to you.

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Definition 1.7

• A sample is a subset of data selected from a
  population.

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Definition 1.8

• A statistical inference is an estimate,
  prediction, or some other generalization about a
  population based on information contained in a
  sample.

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Definition 1.9

• A measure of reliability is a statement (usually
  quantified with a probability value) about the
  degree of uncertainty associated with a
  statistical inference.

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Definition 1.10

• A representative sample exhibits characteristics
  typical if those possessed by the population.

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Definition 1.11

• A random sample of n experimental units is one
  selected from the population in such a way that
  every different sample of size n has an equal
  probability (chance) of selection.

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Describing Quantitative Data Numerically
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Definition 1.15

• The mean of a sample of n measurements is

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Notation

• Sample mean
• Population mean

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Definition 1.16

• The range of a sample of n measurements is
  the difference between the largest and smallest
  measurements in the sample.

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Definition 1.17

• The variance of a sample of n measurements
  is defined to be

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Definition 1.18

• The standard deviation of a set of measurements
  is equal to the square root of their variance.
  Thus, the standard deviation of a sample and a
  population areSample standard deviation
  sPopulation standard deviation

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Guidelines for Interpreting a Standard Deviation

• For any data set (population or sample), at least
  three-fourths of the measurements will lie within
  2 standard deviations of their mean.
• For most data sets of moderate (say, 25 or more
  measurements) with a mound-shaped distribution,
  approximately 95 of the measurements will lie
  within 2 standard deviations of their mean.

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Definition 1.19

• Numerical descriptive measures of a population
  are called parameters.

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Definition 1.20

• A sample statistic is a quantity calculated from
  the observations in a sample.

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Standard normal distribution
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Definition 1.21

• The sampling distribution of a sample statistic
  calculated from a sample of n measurements is the
  probability distribution of the statistic.

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Theorem 1.1

• If represent a random sample of n
  measurements for a large (or infinite) population
  with mean and standard deviation then,
  regardless of the form of the population relative
  frequency distribution, the mean and standard
  error of estimate of the sampling distribution of
  will beMean Standard error of estimate

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The Central Limit Theorem

• For large sample sizes, the mean of a sample
  from a population with mean and standard
  deviation has a sampling distribution that is
  approximately normal, regardless of the
  probability distribution of the sampled
  population. The larger sample size, the better
  will be the normal approximation to the sampling
  distribution of

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Estimating a Population Mean

• If the mean of the sampling distribution of a
  statistics equals the parameter we are
  estimating, we say that the statistic is an
  unbiased estimator of the parameter. If not, we
  say that it is biased.

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Fig. 1.17 Sampling distribution of

• See Applet
• (http//www.ruf.rice.edu/lane/stat_sim/sampling_
  dist/index.html)

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Large-Sample 100(1-?) Confidence Interval for ?

• where is the z value with and area ?/2 to
  its right (see Figure 1.18) and The parameter
  ? is the standard deviation of the sampled
  population and n is the sample size. If ? is
  unknown, its value may be approximated by the
  sample deviation s. The approximation us valid
  for large samples (e.g., n ? 30) only.

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Small-Sample Confidence Interval for ?

• where and is a t value based on (n 1)
  degrees of freedom, such that the probability
  that is ?/2.
• Assumptions The relative frequency distribution
  of the sampled population is approximately normal.

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Testing a Hypothesis About a Population Mean

• A null hypothesis, denoted by the symbol which
  is the hypothesis that we postulate is true
• An alternative (or research) hypothesis, denoted
  by the symbol which is counter to the null
  hypothesis and is what we want to support.

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• A test statistic, calculated from the sample
  data, that functions as a decision maker.
• A rejection region, values of a test statistic
  for which we reject the null hypothesis and
  accept the alterative hypothesis.

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Large-Sample (n?30) Test of Hypothesis About ?

• TWO-TAILED TEST
• Test statistic
• Rejection region
• where is chosen so that

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Type I and Type II errorSmall-Sample Test of
Hypothesis About ?

• TWO-TAILED TEST
• Test statistic
• Rejection region
• where is based on (n 1) df

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Reporting Test Results as p-Values How to Decide
Whether to Reject H0

• Choose the maximum value of ? that you are
  willing to tolerate.
• If the observed significance level (p-value) of
  the test is less than the maximum value of ?,
  then reject the null hypothesis.

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Large-Sample Confidence Interval for (m1-m2)
Independent Samples

• Assumptions The two samples are randomly and
  independently selected from the two populations.
  The sample sizes, n1 and n2, are large enough so
  that and each have approximately normal
  sampling distributions and so that and
  provide good approximations to and This
  will be true if n1 ? 30 and n2 ? 30.

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Large-Sample Test of Hypothesis About (m1-m2)
Independent Samples

• TWO-TAILED TEST
• where D0Hypothesized difference between the
  means (this is often 0)
• Test statistic
• where Rejection region

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Small-Sample Confidence Interval for (m1-m2)
Independent Samples

• where
• Is a pooled estimate of the common population
  variance and ta/2 is based on (n1n2-2) df.
• Assumptions
• Both sampled populations have relative frequency
  distributions that are approximately normal.
• The population variances are equal.
• The samples are randomly and independently
  selected from the populations.

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Small-Sample Test of Hypothesis About (?1-?2)
Independent Samples

• TWO-TAILED TEST
• Test statistic
• Rejection region
• where t? is based on (n1n2-2)df
• Assumptions Same as for the small-sample
  confidence
• interval for (?1-?2) in the previous box.

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Paired Difference Confidence Interval for mdm1-m2

• LARGE SAMPLE
• Assumption The sample differences are randomly
  selected from the population of differences.

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Continued

• SMALL SAMPLE
• where ta/2 is based on (nd-1) degrees of freedom
• Assumptions
• The relative frequency distribution of the
  population of differences is normal.
• The sample differences are randomly selected from
  the population of differences.

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Paired Difference Test of Hypothesis for mDm1-m2

• TWO-TAILED TEST
• LARGE SAMPLE
• Test statistic
• Rejection region
• Assumption The differences are randomly selected
  from the population of differences.

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Continued

• SMALL SAMPLE
• Test statistic
• Rejection region
• where ta/2 is based on (nd-1) degrees of freedom
• Assumptions
• The relative frequency distribution of the
  population of differences is normal.
• The differences are randomly selected from the
  population of differences.