Inferential Statistics

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Inferential Statistics Hypothesis Testing

• Heibatollah Baghi, and
• Mastee Badii

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Objectives

• Conduct one sample mean test
• Using Z statistics
• Using t statistics

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Inferential Statistics Usage

• Researchers use inferential statistics to address
  two broad goals
• Estimate the value of population parameters
• Hypothesis testing

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Distribution of Coin Tosses
If you see 10 heads in a row, is it a fair coin?
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Sample Population

• Think of any sequence of throws as a sample from
  all possible throws
• Think of all possible throws as the entire
  population.
• One-Sample Inferential Tests estimate the
  probability that a sample is representative of
  the total population (within /- 2 standard
  deviations of the mean, or the middle 95 of the
  distribution).

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Logic of Hypothesis Testing

• Is the value observed consistent with the
  expected distribution?
• On average, 100 coin tosses should lead to 50/50
  chance of heads.
• Some coin tosses will be outliers, giving
  significantly different results.
• Are differences significant or merely random
  variations?

Statistics is the art of making sense of
distributions
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Logic of Hypothesis Testing

• The further the observed value is from the mean
  of the expected distribution, the more
  significant the difference

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What about this point?
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What about this point?
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Is this point part of the distribution?
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Probability of Membership in a Distribution

• Depends on location
• Mean
• Variance
• It is a chance event

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One-Sample Tests

• We set a standard beyond which results would be
  rare (outside the expected sampling error)
• We observe a sample and infer information about
  the population
• If the observation is outside the standard, we
  reject the hypothesis that the sample is
  representative of the population

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Random Sampling

• A simple random sampling procedure is one in
  which every possible sample of n objects is
  equally likely to be chosen.
• The principle of randomness in the selection of
  the sample members provides some protection
  against the sample unrepresentative of the
  population.
• If the population were repeatedly sampled in this
  fashion, no particular subgroup would be over
  represented in the sample.

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Sampling Distribution

• The concept of a sampling distribution, allows us
  to determine the probability that the particular
  sample obtained will be unrepresentative.
• On the basis of sample information, we can make
  inference about the parent population.

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Sampling Distribution

• Sampling Error.
• No sample will have the exact same mean and
  standard deviation as the population
• Sampling distribution of the mean
• In research sampling error is often unknown since
  we do not have the population parameters
• A distribution of means of several different
  samples of our population
• Less widely distributed than the population
• Usually Normal

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Population of IQ scores, 10-year olds
µ100 s16
n 64
Sample 2
Sample1
Sample 3
Etc
Is sample 2 a likely representation of our
population?
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Distribution of Sample Means

• The mean of a sampling distribution is identical
  to mean of raw scores in the population (µ)
• If the population is Normal, the distribution of
  sample means is also Normal
• If the population is not Normal, the distribution
  of sample means approaches Normal distribution as
  the size of sample on which it is based gets
  larger

Central Limit Theorem
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Standard Error of the Mean

• The standard deviation of means in a sampling
  distribution is known as the standard error of
  the mean.
• It can be calculated from the standard deviation
  of observations
• The larger our sample size, the smaller our
  standard error

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Sample of observations
Entire population of observations
Random selection
Parameter µ?
Statistic
Statistical inference
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Estimation Procedures

• Point estimates
• For example mean of a sample of 25 patients
• No information regarding probability of accuracy
• Interval estimates
• Estimate a range of values that is likely
• Confidence interval between two limit values
• The degree of confidence depends on the
  probability of including the population mean?

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When Sample size is small
A constant from Student t Distribution that
depends on confidence interval and sample size
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HYPOTHESIS TESTING

• Hygiene procedures are effective in preventing
  cold.
• State 2 hypotheses
• Null H0 Hand-washing has no effect on bacteria
  counts.
• Alternative Ha Hand-washing reduces bacteria.
• The null hypothesis is assumed true i.e., the
  defendant is assumed to be innocent.

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TWO TYPES OF ERROR
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Alpha Beta Errors
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Two Types of Error in Admission to ICU

• Correct decisions
• Patients admitted to ICU who would have failed if
  otherwise
• Patients denied admission who do fine in step
  down unit
• Errors
• Patient admitted who does not need to be there
• Patient denied admission who needs to be there

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Two Types of Error

• Alpha a
• Probability of Type I Error
• P (Rejecting Ho when Ho is true)
• Beta ß
• Probability of Type II Error
• P (Failing to reject Ho when Ho is false)

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Power Confidence Level

• Power
• 1- ß
• Probability of rejecting Ho when Ho is false
• Confidence level
• 1- a
• Probability of failing to reject Ho when Ho is
  true

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Steps in Test of Hypothesis

• Determine the appropriate test
• Establish the level of significancea
• Determine whether to use a one tail or two tail
  test
• Calculate the test statistic
• Determine the degree of freedom
• Compare computed test statistic against a tabled
  value

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1. Determine Appropriate Test

• Level of measurement
• Number of groups being compared
• Sample size
• Extent to which assumption for parametric tests
  have been met
• Relatively Normal distribution
• Approximately interval level variable

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2. Establish Level of Significance

• a is a predetermined value
• The convention
• a .05
• a .01
• a .01

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3. Determine Whether to Use One or Two Tailed Test

• If the alternative hypothesis specifies direction
  of the test, then one tailed
• Otherwise, two tailed
• Most cases

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4. Calculating Test Statistics

• For one sample tests, use Z test statistic if
  population is Normal, ? is known, or if sample
  size is large
• For one sample tests, use T static if population
  distribution is not known or if sample size is
  small (less than 30)

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5. Determine Degrees of Freedom

• Number of components that are free to vary about
  a parameter
• Df Sample size Number of parameters estimated
• Df is n-1 for one sample test of mean

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6. Compare the Computed Test Statistic Against a
Tabled Value
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Example of Testing Statistical Hypotheses About µ
When s is Known(Large Sample Test for Population
Mean).
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Research Question

• Does Home Schooling Affect Educational Outcomes?

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Statistical Hypotheses

• Dr. Tate, a researcher at GMU decided to conduct
  a study to explore this question. He found out
  that every fourth-grade student attending school
  in Virginia takes CAT.
• Scores of CAT are normally distributed with µ
  250 and s 50.
• Home schooled children are not required to take
  this test.

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Statistical Hypotheses

• Dr. Tate selects a random sample of 36 home
  schooled fourth graders and has each child
  complete the test. (It would be too expensive and
  time-consuming to test the entire population of
  home-schooled fourth-grade students in the sate.)
• Step 1 Specify Hypotheses
• H0 µ 250
• Ha µ gt 250
• a 0.05

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Calculated Z

• Select the sample, calculate the necessary sample
  statistics
• n36
• s 50

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Critical Z

• Determine za
• ? 0.05 one sided
• CI of 95
• Refer to the Z table and find the corresponding Z
  score
• Z 1.65

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Make Decisions Regarding Ho

• Because the calculated z is greater than the
  critical z, Ho is rejected.
• 1.80 gt 1.65 and Ha is accepted
• The mean of the population of home-school fourth
  graders is not 250.

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Alternative Steps

• Step 1 Specify Hypotheses
• Ho µ 250 Ha µ gt 250 a .05
• Step 2 Select the sample, calculate sample
  statistics n36 s 50

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Using P value to Reject Hypothesis

• Step 3 Determine the p-value . A z of 1.80
  corresponds to a one tailed probability of
  0.036.
• Step 4 Make decision regarding Ho. Because the
  p-value of 0.036 is less than a 0.05
• H0 is rejected. The mean of the population of
  home-school fourth graders is not 250.

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DECISION RULES

• In terms of z scores
• If Zc gt Za Reject H0
• In terms of p-value
• If p value lt a Reject H0

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The One-sample Z Test

• One-Sample tests of significance are used to
  compare a sample mean to a (hypothesized)
  population mean and determine how likely it is
  that the sample came from that population. We
  will determine the extent to which they occur by
  chance.
• We will compare the probability associated with
  our statistical results (i.e. probability of
  chance) with a predetermined alpha level.

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The One-sample Z Test

• If the probability is equal to or less than our
  alpha level, we will reject the null hypothesis
  and conclude that the difference is not due to
  chance.
• If the probability of chance is greater than our
  alpha level, we will retain the null hypothesis
  and conclude that difference is due to chance.

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Take Home Lesson

• Procedures for Hypothesis Testing and Use of
  These Procedures in One Sample Mean Test for
  Normal Distribution