The t Test for a Single Sample Mean

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The t Test for a Single Sample Mean

• Chapter 11
• SPSS Holcomb
• EDLD 6333 F08

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t Test for a Single Sample Mean

• Learning Objectives
• Learn purpose of t test for a single sample mean
• Discover particulars of the test
• Understand definitions related to test
• Discover how to conduct a t test
• To determine significance of the difference b/w a
  single sample mean a test value

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t Test cont.

• Learning Objectives cont.
• Understand how to interpret results of the t test
• In terms of the null hypothesis
• In terms of statistical significance
• Learn how to present results of t test in APA
  report

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Purpose of t Test for a Single Sample Mean

• Purpose of Test
• To determine if mean for a random sample of
  participants differs significantly from a known
  value or a hypothetical value
• Reminder Mean is the average (as in mathematical
  average) must be used with interval/ratio
  variables

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Purpose of t Test for a Single Sample Mean cont.

• Particulars of Test
• Can only use interval/ratio (scale) data
• Reminder Interval/ratio is a measurement
• Can only use for distributions not highly skewed
• A single sample 1 sample

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Definitions for t Test for a Single Sample Mean

• Definitions of Test
• Known value is a value (score) already determined
• Test Value other term for Hypothetical value in
  SPSS
• Hypothetical value is a assertion or conjecture
  used for test, a value derived from theory
• Test Value other term for Hypothetical value in
  SPSS
• Single sample mean is what researcher will find
• The value (score) by using statistical analysis
  (mean)

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Definitions for t Test cont.

• Definitions of Test cont.
• Significantly different
• Researchers use statistical procedures to
  determine significance of difference
• Discrimination b/w 2 statistical hypotheses
• Null hypothesis
• Alternative hypothesis (more to come )

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Example of t Test for a Single Sample Mean

• Example of a sample mean compared w/ a known
  value
• Researcher knows national average on the ABC
  Attitude Toward Math Scale is 4.0 Researcher
  draws a random sample of students from an urban
  school district, gives attitude scale to the
  sample, obtains a mean of 3.7. To determine if
  the sample mean of 3.7 is significantly different
  from the national average of 4.0, researcher
  needs to conduct a t test for a single sample.

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Example of t Test for a Single Sample Mean cont.

• Example cont.
• Researcher knows national average on the ABC
  Attitude Toward Math Scale is 4.0 Researcher
  draws a random sample of students from an urban
  school district, gives attitude scale to the
  sample, obtains a mean of 3.7. To determine if
  the sample mean of 3.7 is significantly different
  from the national average of 4.0, researcher
  needs to conduct a t test for a single sample.

Known value
Single sample
aka Test Value
Mean (average) of Single Sample
Statistical procedure to determine significance
Research Hypothesis
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Conducting a t Test for a Single Sample Mean

• SPSS Procedures for t test Single Sample
• Open SPSS data file or
• Create new data file
• In Variable View
• Click Analyze
• Click Compare Means
• Click One-Sample T Test

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Conducting a t Test cont.

• SPSS Procedures cont.
• In One-Sample T Test box
• Click on variable want to use for test
• Click on arrowhead
• Moves chosen variable to Test Variable(s) box
• Type in number from known or hypothetical
  value in Test Value box at bottom
• Click Ok

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Interpreting SPSS Output for t Test

• Interpreting Output

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Interpreting Output for t Test cont.
Sample size
Aka SD of sampling distribution

• Interpreting Output cont.

Standard Deviation
Mean
Single Sample variable
Standard Error of the Mean
Reminder SD is the of score points out from
the Mean of a normal distribution
Amount of variability across sample from same
population
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Interpreting Output for t Test cont.
Known value aka Test value

• Interpreting Output cont.

Difference b/w means
df sample size minus 1
95 certain true difference b/w means in this
range
Degrees of freedom
Variable Name
t score
This tests hypothesis
t not far enough out in tail to show significance
here
Significance aka Probability value (p)
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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Standard Error of the Mean cont.
• How much variability across sample from same
  population
• Larger values may indicated sample not
  representative of population statistic came from

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Standard Deviation (reminder)
• A Measure of Variability
• Reminder Variability is the extent to which
  scores vary/differ from each other
• Ex all participants have a score of 10 then no
  variability and the SD will equal 0.00

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Standard Deviation (reminder) cont.
• Number of score points out from the Mean of a
  normal distribution that includes 34 of the
  cases
• SD will increase in size as scores differ from
  each other

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Degrees of Freedom df
• Simple version N-1 (sample size minus one)
• Extended version of individual pieces of info
  utilized in calculating a statistic (estimating a
  parameter) minus the one being estimated
• Usually N-1, b/c the one thing being estimated
  is taken away from the N
• Reminder Parameter is an unknown quantitative
  value used to represent a certain population
  characteristic

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• 95 Confidence Interval of the Difference
• A 95 probability (chance-95 sure) that the
  ACTUAL difference is in there
• Based on the sample difference, sample size,
  sample variability
• Lower upper values of range represent low
  high points where the REAL difference exists

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• 95 CI of the Difference cont.
• It's "inferential - inferring TO the population
  from the sample
• Not 100 certain that the TRUE difference
  (between two means) is in the interval, but we
  are 95 certain
• CI of the Difference is also another way to show
  if p is significant or not

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• 95 CI of the Difference cont.
• If CI of the Difference is both positive
  negative then p is no significance
• If zero (0) is in CI of the Difference range then
  p is no significance
• If CI of the Difference is both positive then
  there is a significance
• If no zero (0) is in CI of the Difference range
  then p is significant

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STATS HUMOR

• Statistics means never having to say you're
  certain.
• http//davidmlane.com/hyperstat/humorf.html

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• t
• A negative value of t indicates
• The sample mean is lower than the Test Value
• If negative then t is on negative side of
  distribution
• A positive value of t indicates
• The sample value is higher than the Test Value
• A t score can also help indicated whether p value
  is significant or not
• t scores can show how far out in tail score is

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• t cont.
• t score like a z score
• If N50 or greater t z score exactly
• If Nlt50 t z scores a little different
• t score is 2 t scores away from zero (0)
• Beyond the 2 its out in the 5

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Sig. (2-tailed)
• SPSS defaults to a 2 tailed t test
• Most common type of t test
• Sig. significance
• Sig. also called the probability value
• p value
• Sig. must be .05 or less to be significant
• Sig. tests the hypothesis

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Guidelines for Probability Values (p Values)
• If p value is equal to or less than .001
• p lt.001
• Declare the difference to be statistically
  significant at the .001 level
• If p value is equal to or less than .01
• p lt.01
• Declare the difference to be statistically
  significant at the .01 level

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Interpreting Output for t Test cont.

• Further Explanation of Terms cont.
• Probability Values cont.
• If p value is equal or less than .05 but greater
  than .01
• p lt.05
• Declare the difference to be statistically
  significant at the .05 level
• If p value is greater than .05
• P gt .05
• Declare the difference to be not statistically
  significant at the .05 level

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STATS HUMOR

• A mathematician, an applied mathematician, and a
  statistician all apply for the same job. At the
  interview, they are asked the question, What is
  11?
• The mathematician replies, "I can prove that it
  exists but not that it is unique."
• The applied mathematician, after some thought,
  replies, "The answer is approximately 1.99, with
  an error in the region of 0.01."
• The statistician steps outside the room, mulls it
  over for several minutes, and eventually returns
  in desperation and inquires, "So what do you want
  it to be?
• http//davidmlane.com/hyperstat/humorf.html

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Hypotheses

• Hypotheses
• Hypothesis
• A (statistical) hypothesis is an assertion or
  conjecture about the distribution of one or more
  random variables
• What you think may be true a prediction
• Null hypothesis
• Assertion hoped to be disproven by data
• Reverse possibility the prediction is wrong
  the predicted effect doesnt exist

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Example of Hypotheses

• Hypothesis example
• If the researcher watches the Washington Redskins
  football team play it will cause the team to
  loose the game
• Null hypothesis example
• The Washington Redskins football team will
  equally win or loose the game regardless of the
  researcher watching the game

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Hypotheses Significance

• Hypothesis Sig
• If p value is significant (p lt .05)
• Reject the null
• Null hypothesis is rejected
• If p value is not significant (p gt .05)
• Failure to reject the null
• Null hypothesis is not rejected

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Interpreting Output for t Test cont.

• Interpreting Output for Example slide 13
• Hypothesis was
• There is a significant statistical difference in
  the scores for the ABC Attitude Toward Math Scale
  between the national average and the XYZ urban
  school district
• Null Hypothesis was
• There is no significant statistical difference in
  the scores for the ABC Attitude Toward Math Scale
  between the national average and the XYZ urban
  school district

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Interpreting Output for t Test cont.

• Interpreting Output for Example slide 13 cont.
• Mean for sample 3.700
• Value of t is -.525
• Probability that the null hypothesis is true is
  .606
• B/c sig. is greater than .05 (p gt .05) difference
  is not statistically significant
• Null hypothesis is not rejected
• 95 CI of Difference is negative positive w/
  zero
• There is no significance

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Presenting Results in APA

• Reporting t Test Results in APA
• Report mean SD for sample
• Report mean SD for known value
• Report value of t
• Indicate whether statistically significant or not
• Use lower case t and italicize
• Include degrees of freedom df
• After reported t in parenthesis

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Presenting Results in APA cont.

• APA t Test Example (Non significant)
• For the local district sample, the values of the
  mean and standard deviation are 3.70 and 2.56,
  respectively. The national mean is 4.00. The
  difference between the sample mean and the
  national mean is not statistically significant at
  the .05 level (t -.525, df 19). Thus, the
  null hypothesis was not rejected.

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Example of Output if Significant
If we ran the same procedure using a Test Value
(Known Value) of 2.50
Range of 95 CI of Difference changed
t is now positive
Test Value changed
p .049 Significant!
Dif. b/w mean is now 1.200
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Presenting Results in APA cont.

• APA t Test Ex. (Significant if test value was
  2.50)
• For the local district sample, the values of the
  mean and standard deviation are 3.70 and 2.56,
  respectively. The national mean is 2.50. The
  difference between the sample mean and the
  national mean is statistically significant at the
  .05 level (t 2.099, df 19). Thus, the null
  hypothesis was rejected.

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Summary

• Researchers use a t test for a single sample mean
  to determine if mean for a random sample of
  participants differs significantly from a known
  value or a hypothetical value
• Only interval/ratio variables are used
• Conducting a t test helps us determine
  significance of the difference b/w a single
  sample mean a test value

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Summary cont.

• Results of the t test help researchers determine
  statistical significance
• Gives Sig. (p value)
• Can either reject the null hypothesis
• Or fail to reject the null
• Results of t test presented in APA report
• Include mean, SD, t, whether statistically
  significant or not, df, if reject null
  hypothesis or fail to reject the null hypothesis