Statistics 222 Review of Statistics 221

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Statistics 222 Review of Statistics 221
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Definitions

• Elements
• Entities on which data is collected
• Variables
• Items of Interest
• Qualitative vs Quantative

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Summary - Levels of Measurement

• Nominal - categories only
• Ordinal - categories with some order
• Interval - differences but no natural
  starting point
• Ratio - differences and a natural starting
  point

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Definitions

• Cross Sectional vs Time Series
• Statistical Inference
• Population vs Sample
• Relative Frequency Frequency of Class / N
• Population vs Sample
• Class Width Largest Data Value Smallest Data
  Value
• Number of classes
• Histogram -gt no gaps
• Ogive -gt Cumulative Frequence
• Scatter Plot

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Best Measure of Center
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Skewness
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Sample Standard Deviation Formula
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Sample Standard Deviation (Shortcut Formula)
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Population Standard Deviation
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Variance - Notation
standard deviation squared
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Standard Deviation from a Frequency Distribution

• Use the class midpoints as the x values

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Estimation of Standard Deviation Range Rule of
Thumb
For estimating a value of the standard deviation
s, Use Where range (highest value) (lowest
value)
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Definition
Empirical (68-95-99.7) Rule For data sets having
a distribution that is approximately bell shaped,
the following properties apply

• About 68 of all values fall within 1 standard
  deviation of the mean

• About 95 of all values fall within 2 standard
  deviations of the mean

• About 99.7 of all values fall within 3 standard
  deviations of the mean

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The Empirical Rule
FIGURE 2-13
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The Empirical Rule
FIGURE 2-13
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The Empirical Rule
FIGURE 2-13
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Definition

• z Score (or standard score) the number of
  standard deviations that a given value x is above
  or below the mean.

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Measures of Position z score

• Sample

Population
Round to 2 decimal places
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Interpreting Z Scores
Whenever a value is less than the mean, its
corresponding z score is negative Ordinary
values z score between 2 and 2 sd Unusual
Values z score lt -2 or z score gt 2 sd
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Quartiles
Q1, Q2, Q3 divides ranked scores into four
equal parts
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Finding the Percentile of a Given Score
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Converting from the kth Percentile to the
Corresponding Data Value
Notation
n total number of values in the data set k
percentile being used L locator that gives the
position of a value Pk kth percentile
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Some Other Statistics

• Interquartile Range (or IQR) Q3 - Q1

• 10 - 90 Percentile Range P90 - P10

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Law of Large Numbers

• As a procedure is repeated again and again,
  the relative frequency probability (from Rule 1)
  of an event tends to approach the actual
  probability.

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Definition
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Graphs
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Requirements for Probability Distribution
0 ? P(x) ? 1 for every individual value of x
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Definition

• Standard Normal Distribution
• a normal probability distribution that has a
• mean of 0 and a standard deviation of 1.

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Finding z Scores when Given Probabilities
5 or 0.05
1.645
(z score will be positive)
Finding the 95th Percentile
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Finding z Scores when Given Probabilities
(One z score will be negative and the other
positive)
Finding the Bottom 2.5 and Upper 2.5
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Nonstandard Normal Distributions

• If ? ? 0 or ?? ? 1 (or both), we will convert
  values to standard scores using Formula 5-2, then
  procedures for working with all normal
  distributions are the same as those for the
  standard normal distribution.

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Converting to Standard Normal Distribution

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Cautions to keep in mind

• 1. Dont confuse z scores and areas.  z scores
  are distances along the horizontal  scale, but
  areas are regions under the  normal curve.
  Table A-2 lists z scores in the left column and
  across the top row, but areas are found in the
  body of the table.
• 2. Choose the correct (right/left) side of the
  graph.
• 3. A z score must be negative whenever it is
  located to the left half of the normal
  distribution.
• 4. Areas (or probabilities) are positive or zero
  values, but they are never negative.
  

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Central Limit Theorem
Given

• 1. The random variable x has a distribution
  (which may or may not be normal) with mean µ and
  standard deviation ?.
• 2. Samples all of the same size n are randomly
  selected from the population of x values.

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Central Limit Theorem
Conclusions
1. The distribution of sample x will, as the
sample size increases, approach a normal
distribution. 2. The mean of the sample means
will be the population mean µ. 3. The standard
deviation of the sample means will approach
??????????????
n

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Practical Rules Commonly Used

• 1. For samples of size n larger than 30, the
  distribution of the sample means can be
  approximated reasonably well by a normal
  distribution. The approximation gets better as
  the sample size n becomes larger.
• 2. If the original population is itself normally
  distributed, then the sample means will be
  normally distributed for any sample size n (not
  just the values of n larger than 30).

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Notation

• the mean of the sample means
• ???????????????

µx µ
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Notation

• the mean of the sample means
• the standard deviation of sample mean
• ???????????????

µx µ
?
?x
n
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Notation

• the mean of the sample means
• the standard deviation of sample mean
• ???
• (often called standard error of the mean)

µx µ
?
?x
n
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DefinitionPoint Estimate

• A point estimate is a single value (or point)
  used to approximate a population parameter.

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DefinitionConfidence Interval

• A confidence interval (or interval estimate)
  is a range (or an interval) of values used to
  estimate the true value of a population
  parameter. A confidence interval is sometimes
  abbreviated as CI.

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DefinitionConfidence Interval

• A confidence level is the probability 1?
  (often expressed as the equivalent
  percentage value) that is the proportion of
  times that the confidence interval actually
  does contain the population parameter,
  assuming that the estimation process is
  repeated a large number of times.

This is usually 90, 95, or 99. (? 10),
(? 5), (? 1)
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The Critical Value
z??2
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Notation for Critical Value
The critical value z?/2 is the positive z value
that is at the vertical boundary separating an
area of ?/2 in the right tail of the standard
normal distribution. (The value of z?/2 is at
the vertical boundary for the area of ?/2 in the
left tail). The subscript ?/2 is simply a
reminder that the z score separates an area of
?/2 in the right tail of the standard normal
distribution.
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Finding z??2 for 95 Degree of Confidence
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Finding z??2 for 95 Degree of Confidence
? 0.05
Use Table to find a z score of 1.96
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Assumptions
1. The sample is a simple random sample. 2.
The value of the population standard deviation ?
is known. 3. Either or both of these conditions
is satisfied The population is normally
distributed or n gt 30.
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Definitions

• Estimator
• is a formula or process for using sample data
  to estimate a population parameter.
• Estimate
• is a specific value or range of values used to
  approximate a population parameter.
• Point Estimate
• is a single value (or point) used to
  approximate a population parameter.

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Sample Mean

• 1. For many populations, the distribution of
  sample means x tends to be more consistent (with
  less variation) than the distributions of other
  sample statistics.
• 2. For all populations, the sample mean x is an
  unbiased estimator of the population mean ?,
  meaning that the distribution of sample means
  tends to center about the value of the population
  mean ?.

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Definition
Level of Confidence

• confidence level is often expressed as
  probability 1 - ?, where ? is the complement of
  the confidence level. For a 0.95(95) confidence
  level, ? 0.05. For a 0.99(99) confidence
  level, ? 0.01.

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Definition
Margin of Error
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Procedure for Constructing a Confidence Interval
for µwhen ? is known

• 1. Verify that the required assumptions are met.

2. Find the critical value z??2 that corresponds
to the desired degree of confidence.
5. Round using the confidence intervals roundoff
rules.
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Sample Size for Estimating Mean ?

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Finding the Sample Size nwhen ? is unknown

• Use the range rule of thumb
• to estimate the standard deviation as follows ?
  ? range/4.

2. Conduct a pilot study by starting the sampling
process. Based on the first collection of at
least 31 randomly selected sample values,
calculate the sample standard deviation s and use
it in place of ?.
3. Estimate the value of ? by using the results
of some other study that was done earlier.
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? Not KnownAssumptions

• 1) The sample is a simple random sample.
• 2) Either the sample is from a normally
  distributed population, or n gt 30.
• Use Student t distribution

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Student t Distribution

• If the distribution of a population is
  essentially normal, then the distribution of

x - µ
t
s
n

• is essentially a Student t Distribution for all
  samples of size n, and is used to find
  critical values denoted by t?/2.

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Margin of Error E for Estimate of ?

• Based on an Unknown ? and a Small Simple Random
  Sample from a Normally Distributed Population

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Confidence Interval for the Estimate of E Based
on an Unknown ? and a Small Simple Random Sample
from a Normally Distributed Population

• t?/2 found in Table t

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Procedure for Constructing a Confidence Interval
for µwhen ? is not known

• 1. Verify that the required assumptions are met.

2. Using n 1 degrees of freedom, refer to Table
A-3 and find the critical value t??2 that
corresponds to the desired degree of confidence.
5. Round the resulting confidence interval limits.
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Student t Distributions for n 3 and n 12
Figure 6-5
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Using the Normal and t Distribution