Testing Statistical Hypothesis The One Sample tTest

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Testing Statistical HypothesisThe One Sample
t-Test

• Heibatollah Baghi, and
• Mastee Badii

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Parametric and Nonparametric Tests

• Parametric tests estimate at least one parameter
  (in t-test it is population mean)
• Usually for normal distributions and when the
  dependent variable is interval/ratio
• Nonparametric tests do not test hypothesis about
  specific population parameters
• Distribution-free tests
• Although appropriate for all levels of
  measurement most frequently applied for nominal
  or ordinal measures

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Parametric and Nonparametric Tests

• Nonparametric tests are easier to compute and
  have less restrictive assumptions
• Parametric tests are much more powerful (less
  likely to have type II error)

What is type two error?
This lecture focuses on One sample t-test which
is a parametric test
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Two Types of Error

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

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Types of Error in Hypothesis Testing Ho
Hand-washing has no effect on bacteria counts
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Types of Error in Hypothesis Testing Ho
Hand-washing has no effect on bacteria counts
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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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Level of Significance

• a is a predetermined value by convention usually
  0.05
• a 0.05 corresponds to the 95 confidence level
• We are accepting the risk that out of 100
  samples, we would reject a true null hypothesis
  five times

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Sampling Distribution Of Means
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Sampling Distribution Of Means

• A sampling distribution of means is the relative
  frequency distribution of the means of all
  possible samples of size n that could be selected
  from the population.

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One Sample Test

• Compares mean of a sample to known population
  mean
• Z-test
• T-test

This lecture focuses on one sample t-test
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The One Sample t Test

• Testing statistical hypothesis about µ when s is
  not known OR sample size is small

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An Example Problem

• Suppose that Dr. Tate learns from a national
  survey that the average undergraduate student in
  the United States spends 6.75 hours each week on
  the Internet composing and reading e-mail,
  exploring the Web and constructing home pages.
  Dr. Tate is interested in knowing how Internet
  use among students at George Mason University
  compares with this national average.
• Dr. Tate randomly selects a sample of only 10
  students. Each student is asked to report the
  number of hours he or she spends on the Internet
  in a typical week during the academic year.

Population mean
Small sample
Population variance is unknown estimated from
sample
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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 the appropriate test

• If sample size is more than 30 use z-test
• If sample size is less than 30 use t-test
• Sample size of 10

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2. Establish Level of Significance

• a is a predetermined value
• The convention
• a .05
• a .01
• a .001
• In this example, assume a 0.05

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3. Determine Whether to Use a One or Two Tailed
Test

• H0 µ 6.75
• Ha µ ? 6.75

A two tailed test because it can be either
larger or smaller
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4. Calculating Test Statistics
Sample mean
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4. Calculating Test Statistics
Deviation from sample mean
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4. Calculating Test Statistics
Squared deviation from sample mean
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4. Calculating Test Statistics

Standard deviation of observations
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4. Calculating Test Statistics

Calculated t value
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4. Calculating Test Statistics

Standard deviation of sample means
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4. Calculating Test Statistics

Calculated t
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5. Determine Degrees of Freedom

• Degrees of freedom, df, is value indicating the
  number of independent pieces of information a
  sample can provide for purposes of statistical
  inference.
• Df Sample size Number of parameters estimated
• Df is n-1 for one sample test of mean because the
  population variance is estimated from the sample

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Degrees of Freedom

• Suppose you have a sample of three observations

X
--------
--------
--------
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Degrees of Freedom

• Why n-1 and not n?
• Are these three deviations independent of one
  another?
• No, if you know that two of the deviation scores
  are -1 and -1, the third deviation score gives
  you no new independent information ---it has to
  be 2 for all three to sum to 0.

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Degrees of Freedom Continued

• For your sample scores, you have only two
  independent pieces of information, or degrees of
  freedom, on which to base your estimates of S and

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6. Compare the Computed Test Statistic Against a
Tabled Value

• a .05
• Df n-1 9
• Therefore, reject H0

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Decision Rule for t-Scores

• If tc gt ta Reject H0

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Decision Rule for P-values

• If p value lt a Reject H0

Pvalue is one minus probability of observing the
t-value calculated from our sample
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Example of Decision Rules

• In terms of t score
• tc 2.449 gt ta 2.262
  Reject H0
• In terms of p-value
• If p value .037 lt a .05 Reject H0

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Constructing a Confidence Interval for µ
Standard deviation of sample means
Sample mean
Critical t value
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Constructing a Confidence Interval for µ for the
Example

• Sample mean is 9.90
• Critical t value is 2.262
• Standard deviation of sample means is 1.29
• 9.90 2.262 1.29
• The estimated interval goes from 6.98 to 12.84

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Distribution of Mean of Samples

• In drawing samples at random, the probability is
  .95 that an interval constructed with the rule
• will include m

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Sample Report of One Sample t-test in Literature
One Sample t-test Testing Neutrality of Attitudes
Towards Infertility Alternatives
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Testing Statistical Hypothesis With SPSS
   
SPSS Output One-Sample Statistics
One-Sample Test

   
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Take Home Lesson

• Procedures for Conducting Interpreting One
  Sample Mean Test with Unknown Variance