Statistics Onesample ttest

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StatisticsOne-sample t-test

• June 20, 2000

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To Do

• Take over world
• Use statistics
• One-sample t-test
• End of chapter 8
• Two-sample and related sample t-tests
• Chapter 9

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• One-sample t-test
• Compared to the z-test
• The t-distribution and df
• Examples of its use
• Confidence intervals
• Power
• What it is
• How to get it

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Moving from the z-test to the t-test

• The z-test may be summarized as follows
• A sample mean is expected to approximate a
  population mu. This allows us to use the sample
  mean to test a hypothesis about the population
  mu.
• The standard error provides a measure of how well
  a sample mean approximates the population mean.
• To quantify our inferences about the population,
  we compare the obtained sample mean with the
  hypothesized population mu by computing a z-score
  test statistic

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Why use the t-test then?

• Usually, we dont know the population standard
  deviation
• Therefore, we cant calculate the standard error
  of the sample means
• Without the standard error of the sample means,
  we dont know how much difference to expect in
  our obtained difference
• However, the t-test allows us to estimate the
  standard error using the sample standard
  deviation
• So the t-test formula looks like the z-test
  formula

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Characteristics of the t-test

• The t statistic is used to test hypotheses about
  a population mu when the population standard
  deviation is not known.
• The t statistic formula is similar to the z
  statistic formula, except the t statistic formula
  uses estimated standard error.
• The estimated standard error is used as an
  estimate of the standard error of the sample mean
  when the population standard deviation is not
  known.
• It is computed from the sample standard deviation
  and provides an estimate of the standard distance
  between a sample mean and the population mu.
• The t distribution is effected by the degrees of
  freedom (sample size)

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Characteristics of the t distribution

• The shape of the t distribution changes by values
  of degrees of freedom (df)
• df describes the number of scores in a sample
  that are free to vary. It is equal to the sample
  size minus one.
• The larger the df (i.e., sample size) the better
  the sample represents its population
• With a large df ( about 120) the t distribution
  is normal (like the z distribution)
• With small df values the t distribution is
  flatter and more spread out
• Not all df values are given in the t table - so
  use the larger critical t value (i.e., for the
  lower df value)

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Critical values from the z-table at different df
values

• The critical values of z for a two-tailed test
  with an ? .05 are -1.96 and 1.96
• For a t with a sample size of 4 and df 3, the
  critical values are -3.182 and 3.182
• For a t with a sample size of 31 and df 30,
  the critical values are -2.042 and 2.042
• For a t with a sample size of 121 and df 120,
  the critical values are -1.980 and 1.980

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Same basic experimental situation
Known population before treatment
Unknown population after treatment
Treatment
? 30
? ?
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Procedures for conducting the t-test

• Step 1 The null and alternative hypotheses are
  stated and the alpha level is set (typically to
  .05)
• Step 2 The critical region is located using the
  df value, direction of the hypothesis, and the
  alpha value in the t distribution table.
• Step 3 The sample data are collected and the test
  statistic is calculated
• Step 4 The null hypothesis is evaluated.

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Conclusions from t results

• If the t statistic falls within the critical
  region (exceeds the value of the critical t) then
  the null hypothesis (Ho) is rejected.
• We then conclude that a treatment effect exists.
• If the t statistic does not fall within the
  critical region (is less than the value of the
  critical t) then we fail to reject the null
  hypothesis (Ho).
• We then conclude that we failed to observe
  evidence for a treatment effect in our study.

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Reporting results of t-tests

• Proper format for reporting t-test results
• t(8) -3.61, p lt .05
• This says
• We preformed the t-test
• We had 8 degrees of freedom
• Our tobt is -3.61
• This probability of our obtained t value is less
  than 5, therefore the difference between the
  mean and the mu is significant and we reject the
  null hypothesis

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Estimating the population mu

• Estimate the population mu using the sample mean
• Point estimation
• Likely to be wrong (too specific)
• Estimate a range of scores that may contain the
  population mu
• Interval estimation
• Accounts for sampling error
• A confidence interval contains the highest and
  lowest values of the population mu that are not
  significantly different from our mean

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Interpreting confidence intervals

• The computational formula for confidence
  intervals is
• Use the two-tailed critical values of t
• The amount of confidence (such as 95) is related
  to the alpha value used when finding tcrit
• Using an alpha of .05, you calculate a 95
  confidence interval
• Using an alpha of .01, you calculate a 99
  confidence interval

(sx)(-tcrit) X ? (sx)(tcrit) X
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Experimental Power

• The power of a statistical test is the
  probability that the test will correctly reject a
  false null hypothesis.
• Power is effected by
• The alpha level, reduce alpha (i.e., from .05 to
  .01) you reduce your power
• One-tailed versus two-tailed tests, one-tailed
  tests have more power
• Sample size, increasing sample size increases
  your power

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Power and Effect Size

• Effect size (i.e., Cohens d) is an indication of
  how much the IV and DV are related
• Hypothesis tests (i.e. the t-test) are an
  indication of how reliable the relationship
  between the IV and DV is
• Power and effect size
• The smaller the effect size, the more power
  required to find the effect
• A larger effect size does not need as much power
• Some non-significant results are due to a lack of
  power

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Two-Sample t-tests

• Transition from one to two sample tests
• Two-sample t-tests
• Independent Samples
• Homogeneity of variance
• Related Samples

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Contrasting the z-test, one-sample and
two-samples t-test

• In the z-test, we know the population mu and
  standard deviation
• In the one-sample t-test, we know the population
  mu but do not know the population standard
  deviation
• estimate the standard error using the sample
  standard deviation
• In the two-samples t-test, we dont know either
  the population mu or standard deviation
• estimate the differences in the population with
  the sample means and estimate the standard error
  of the differences using the pooled sample
  variances

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Comparing the z-test, one-sample and two-samples
t-tests

• The test equation has the same basic form

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Two-sample research design

• An experiment that uses a separate sample for
  each treatment condition (or each population) is
  call an independent-samples research design
• We are interested in the mean difference between
  two populations

Population A
Population B
Unknown ? ?
Unknown ? ?
?A - ?B gt 0
Sample A
Sample B
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Assumptions of the two-sample t-test

• The two random samples of dependant scores
  measure an interval or ratio variable
• The population of raw scores represented by each
  sample is normally distributed and best described
  by the mean.
• We do not know the variance of either population
  and must estimate it from the sample data.
• The populations represented by our samples have
  homogeneous variance.
• Homogeneity of variance means that the true
  variance of the two population distributions are
  the same.
• This is especially important when the sample
  sizes are not equal.

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Null and alternative hypotheses tested by the
two-sample t-test

• Two-tailed hypotheses predicting a mean
  difference of zero
• Ha ?1 - ?2 ? 0
• Ho ?1 - ?2 0
• Two-tailed hypotheses predicting a non-zero mean
  difference
• Ha ?1 - ?2 ? 10
• Ho ?1 - ?2 10
• One-tailed hypotheses predicting a difference of
  zero
• Ha ?1 - ?2 gt 0
• Ho ?1 - ?2 0

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Steps in conducting a independent-samples t-test

• Identify your experimental hypothesis (then your
  Ho and Ha) and select an alpha level (typically
  .05)
• Collect data from samples that meet the
  assumptions of the two-sample t-test and
  calculate the means and variances
• Calculate the pooled variance using the sample
  variances then calculate the standard error of
  the difference using the pooled variance then
  calculate tobt using the standard error of the
  difference
• Compare your tobt with the tcrit in the tables
• For two-sample t-tests, df (n1 - 1) (n2-1)
• Report your results and graph the means
• Calculate the confidence interval for the mean
  differences
• Calculate the effect size (using either rpb or d)

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Calculating the error term of the two-sample
t-test

• In estimating two population mus we have two
  sources of error
• x1 approximates ?1 with some error
• x2 approximates ?2 with some error
• We are interested in the combined error of the
  samples so we pool the variance
• This gives us an average error of the two
  samples
• Weigh variances by their df
• Way of accounting for differences in sample sizes
• Larger samples get more weight because they are
  better estimates of the population
• Use the pooled variance to calculate the standard
  error of the difference

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Sampling distribution of mean differences when ?1
- ?1 0

• We are interested in the significance of the
  difference of our sample scores
• So we have a sampling distribution of mean
  differences
• We are comparing our differences in sample means
  to the differences in population mus given by the
  null hypothesis

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Results of the t-test

• Present the results of your t-test
• t(30) 2.94, p lt .05
• df 30
• tobt 2.94
• Difference is significant
• Calculate the confidence interval of the mu
  difference
• If we preformed the experiment on the population,
  we are 95 confident that the difference would be
  between about .90 and 5.08
• Calculate the effect size
• The strength of the relationship or how much the
  independent and dependant variable are related
• rpb2 (.10, .30, .50)
• d tobt (.20, .50, .80)

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Power

• How to enhance the power of you two-sample t-test
• Maximize the difference produced by the two
  conditions
• High impact manipulations
• Very different conditions of the independent
  variable
• Minimize the variability of the raw scores
• Good experimental control
• Eliminate extraneous variables
• Maximize the sample ns
• Smaller denominator when calculating tobt
• Larger df resulting in a smaller value of tcrit

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Related-sample t-test

• t-Test experiments
• Related-sample t-test
• Designs
• Dependent variable
• Pros and cons

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Types of designs using t-tests

• Single-sample
• One sample of subjects
• Comparing the mean and the mu
• Independent-samples
• Two samples of subjects
• Comparing two means
• Repeated-measures
• One sample of subjects measured twice
• Looking at the difference between the means of
  the two measurements
• Matched-subjects
• Two samples of subjects that are paired on a
  certain variable
• Looking at the difference between the two means

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Designs for related samples

• Matched-sample design
• Each individual in one sample is matched with a
  subject in the other sample
• The matching is done so that the two individuals
  are equivalent (or nearly equivalent) with
  respect to a specific variable that the
  researcher would like to control
• Repeated-measures design
• A single sample of subjects are used to compare
  two different treatment conditions
• Each individual is measured in one treatment, and
  then the same individual is measured again in the
  second treatment. Thus, a repeated-measures study
  produces two sets of scores, but each is obtained
  from the same sample of subjects

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Pros and Cons of Related Samples Designs

• Matched-samples design
• Pro More powerful control individual
  differences
• Con Matched on wrong variable
• Repeated-measures design
• Pro Even more powerful better control of
  individual differences
• Con Order effects
• The first survey may effect performance on the
  second survey
• Counter balance 50 get A then B 50 get B then
  A

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Dependent variable in related-samples t-tests

• In both cases (matched and repeated designs), we
  subtract one score from the other and do a
  one-sample-like t-test on the average difference
  (D)
• Instead of the mean of x, we use the mean of D
• D x2 - x1 after - before
• Instead of a known mu value, we use a value given
  in the null hypothesis (i.e., set by the
  experimenter)

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Hypotheses Tested

• Interested in whether or not any difference
  exists between scores in the first treatment and
  scores in the second treatment.
• Is the population mean difference (?D) equal to
  zero (no change) or has a change occurred?
• Two-tailed hypotheses
• Ha ?D ? 0 or Ha ?D ? 20
• Ho ?D 0 Ho ?D 20
• One-tailed hypotheses
• Ha ?D gt 0
• Ho ?D 0

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Results of the t-test

• Present the results of your t-test
• t(30) 2.94, p lt .05
• df 30
• tobt 2.94
• Difference is significant
• Calculate the confidence interval of the mu
  difference
• If we preformed the experiment on the population,
  we are 95 confident that the difference would be
  between about .90 and 5.08

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Effect Size

• Calculate the effect size
• The strength of the relationship or how much the
  independent and dependant variable are related
• rpb
• Small - 10
• Medium - 30
• Large - 50
• d tobt
• Small - 20
• Medium - 50
• Large - 80

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Power

• How to enhance the power of your two-sample
  t-test
• Maximize the difference produced by the two
  conditions
• High impact manipulations
• Very different conditions of the independent
  variable
• Minimize the variability of the raw scores
• Good experimental control
• Eliminate extraneous variables
• Maximize the sample ns
• Smaller denominator when calculating tobt
• Larger df resulting in a smaller value of tcrit

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Learning Check

• 1 What assumptions must be satisfied for the
  repeated-measures t-test to be valid?
• Random samples, interval or ratio variables,
  population of differences (D) is normally
  distributed, homogeneous variance (always equal n
  sizes)
• 2 Describe some situations for which a
  repeated-measure design is well suited.
• When subjects are hard to find (requires less
  subjects) required by your research question
  (differences over time) individual differences
  are large (reduces error)

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Learning Check

• 3 How is a matched-subjects design similar to a
  repeated-measures design? How do they differ?
• They both reduce the role of individual
  differences thereby increasing power. They differ
  in that there are two samples in the matched
  design and one in the repeated measures design.
• 4 The data from a research study consist of 8
  scores for each of two different treatment
  conditions. How many individual subjects would be
  needed to produce these data.
• a. For an independent-measures design?
• 16, two separate sample with n 8 in each
• b. For a repeated measures design?
• 8 subjects, the same 8 subjects are measured in
  both treatments
• c. For a matched-subjects design?
• 16 subjects, 8 matched pairs

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The End