Stats 95 t-Tests

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Stats 95t-Tests

• Single Sample
• Paired Samples
• Independent Samples

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t Distributions

• t dist. are used when we know the mean of the
  population but not the SD of the population from
  which our sample is drawn
• t dist. are useful when we have small samples.
• t dist is flatter and has fatter tails
• As sample size approaches 30, t looks like z
  (normal) dist.

• Same Three Assumptions
• Dependent Variable is scale
• Random selection
• Normal Distribution

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Fat Tails Lose Weight With Larger Sample Size
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The Robust Nature of the t Statistics

• Unfortunately, we very seldom know the if the
  population is normal because usually all the
  information we have about a population is in our
  study, a sample of 10-20.
• Fortunately,
• 1) distributions in social sciences often
  approximate a normal curve, and
• 2) according to Central Limit Theorem the sample
  mean you have gathered is part of a normal
  distribution of sample means, and
• 3) in practice t tests statisticians have found
  the test is accurate even with populations far
  from normal

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The Robust Nature of the t Statistics

• The only situation in which using a t test is
  likely to give a seriously distorted result is
  when you are using a one-tailed test and the
  population is highly skewed.

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z Statistic Versus t Statistic

• z Statistic

• t Statistic

• When you know the Mean and Standard deviation of
  a population.
• E.g., a farmer picks 200,000 apples, the mean
  weight is 112 grams, the SD is 12grams.
• Calculate the Standard Error of the sample mean

• When you do not know the Mean and Standard
  Deviation of the population
• E.g., a farmer picks 30 out of his 200,000
  apples, and finds the sample has a Mean of 112
  grams.
• Calculate the Estimate of the Standard Error of
  the sample mean

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Scenarios When you would use a Single Sample t
test

• A newspaper article reported that the typical
  American family spent an average of 81 for
  Halloween candy and costumes last year. A sample
  of N 16 families this year reported spending a
  mean of M 85, with s 20. What statistical
  test would we use to determine whether these data
  indicate a significant change in holiday
  spending?
• Many companies that manufacture lightbulbs
  advertise their 60-watt bulbs as having an
  average life of 1000 hours. A cynical consumer
  bought 30 bulbs and burned them until they
  failed. He found that they burned for an average
  of M 1233, with a standard deviation of s
  232.06. What statistical test would this consumer
  use to determine whether the average burn time of
  lightbulbs differs significantly from that
  advertised?

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Difference Between Calculating z Statistic and t
Statistic

• z Statistic

• t Statistic

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Estimating Population from a Sample

• Main difference between t Tests and z score
• use the standard deviation of the sample to
  estimate the standard deviation of the
  population.
• How? Subtract 1 from sample size! (called degrees
  of freedom)
• Use degrees of freedom (df) in the t distribution
  chart

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t Distribution Table
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Example of Single Sample t Test

• The mean emission of all engines of a new design
  needs to be below 20ppm if the design is to meet
  new emission requirements. Ten engines are
  manufactured for testing purposes, and the
  emission level of each is determined. Data
• 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9,
  12.7, 13.9
• Does the data supply sufficient evidence to
  conclude that type of engine meets the new
  standard, assuming we are willing to risk a Type
  I error (false alarm, reject the Null when it is
  true)) with a probability 0.01?
• Step 1 Assumptions dependent variable is scale,
  Randomization, Normal Distribution
• Step 2 State H0 and H1
• H0 Emissions are equal to (or greater than)
  20ppm
• H1 Emissions are lesser than 20ppm
  (One-Tailed Test)

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Example of Single Sample t Test

• The mean emission of all engines of a new design
  needs to be below 20ppm if the design is to meet
  new emission requirements. Ten engines are
  manufactured for testing purposes, and the
  emission level of each is determined. Data
• 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9,
  12.7, 13.9
• Step 3 Determine Characteristics of Sample
• Mean
• Standard Deviation of Sample
• Standard Error of Sample
• Step 4 Determine Cutoff
• df N-1 10-1 9
• t statistic cut-off -2.822

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Example of Single Sample t Test

• The mean emission of all engines of a new design
  needs to be below 20ppm if the design is to meet
  new emission requirements. Ten engines are
  manufactured for testing purposes, and the
  emission level of each is determined. Data
• 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9,
  12.7, 13.9
• Step 3 Determine Characteristics of Sample
• Mean M
  17.17
• Standard Deviation of Sample
  s 2.98
• Standard Error of Sample sm
  0.942
• Step 4 Determine Cutoff
• df N-1 10-1 9
• t statistic cut-off -2.822

Step 5 Calculate t Statistic
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Example of Single Sample t Test

• The mean emission of all engines of a new design
  needs to be below 20ppm if the design is to meet
  new emission requirements. Ten engines are
  manufactured for testing purposes, and the
  emission level of each is determined. Data
• 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9,
  12.7, 13.9
• Mean M 17.17 Standard Deviation
  of Sample s 2.98 Standard Error
  of Sample sm 0.942

Step 5 Calculate t Statistic
Step 6 Decide (Draw It) t statistic cut-off
-2.822 t statistic -3.00 Decide to reject
the Null Hypothesis
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Paired Sample t Test

• The paired samples test is a kind of research
  called repeated measures test (aka,
  within-subjects design), commonly used in
  before-after-designs.
• Comparing a mean of difference scores to a
  distribution of means of difference scores
• Population of measures at Time 1 and Time 2
• Population of difference between measures at Time
  1 and Time 2
• Population of mean difference between measures
  at Time 1 and Time 2
• (Whew!)

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Paired Sample t Test

• Single-Sample

• Paired-Sample

• Single observation from each participant
• The observation is independent from that of the
  other participants
• Comparing a mean score to a distribution of mean
  scores .

• Two observations from each participant
• The second observation is dependent upon the
  first since they come from the same person.
• Comparing a mean of difference scores to a
  distribution of means of difference scores
• (I dont make this stuff up)

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Paired Sample t Test

• A distribution of scores.

• A distribution of differences between scores.

• Central Limit Theorem Revisited. If you plot the
  mean of randomly sampled observations, the plot
  will approach a normal distribution. This is true
  for scores and for differences between scores.

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Difference Between Calculating Single-Sample t
and Paired-Sample t Statistic

• Single Sample t Statistic

• Paired Sample t Statistic

Standard Deviation of Sample Differences
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Paired Sample t Test Example

• We need to know if there is a difference in the
  salary for the same job in Boise, ID, and LA, CA.
  The salary of 6 employees in the 25th percentile
  in the two cities is given .
• Six Steps of Hypothesis testing for Paired Sample
  Test

Profession Boise Los Angeles
Executive Chef 53,047 62,490
Genetics Counselor 49,958 58,850
Grants Writer 41,974 49,445
Librarian 44,366 52,263
School teacher 40,470 47,674
Social Worker 36,963 43,542
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Paired Sample t Test Example

• We need to know if there is a difference in the
  salary for the same job in Boise, ID, and LA, CA.
  
• Step 1 Define Pops. Distribution and Comparison
  Distribution and Assumptions
• Pop. 1. Jobs in Boise
• Pop. 2.. Jobs in LA
• Comparison distribution will be a distribution
  of mean differences, it will be a paired-samples
  test because every job sampled contributes two
  scores, one in each condition.
• Assumptions the dependent variable is scale, we
  do not know if the distribution is normal, we
  must proceed with caution the jobs are not
  randomly selected, so we must proceed with caution

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Paired Sample t Test Example

• We need to know if there is a difference in the
  salary for the same job in Boise, ID, and LA, CA.
  
• Step 3 Determine the Characteristics of
  Comparison Distribution (mean, standard
  deviation, standard error)
• M 7914.333 Sum of Squares (SS)
  5,777,187.333

Profession Boise Los Angeles X-Y D (X-Y)-M M 7914.33 D2
Executive Chef 53,047 62,490 -9,443 -1,528.67 2,336,821.78
Genetic Counselor 49,958 58,850 -8,892 -977.67 955,832.11
Grants Writer 41,974 49,445 -7,471 443.33 196,544.44
Librarian 44,366 52,263 -7,897 17.33 300.44
School teacher 40,470 47,674 -7,204 710.33 504,573.44
Social Worker 36,963 43,542 -6,579 1,335.33 1,783,115.11
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Paired Sample t Test Example

• We need to know if there is a difference in the
  salary for the same job in Boise, ID, and LA, CA.
  
• Step 4 Determine Critical Cutoff
• df N-1 6-1 5
• t statistic for 5 df , p lt .05, two-tailed, are
  -2.571 and 2.571
• Step 5 Calculate t Statistic
• Step 6 Decide

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Independent t Test

• Compares the difference between two means of two
  independent groups.
• The comparison distribution is a difference
  between means to a distribution of differences
  between means.
• Population of measures for Group 1 and Group 2
• Sample means from Group 1 and Group 2
• Population of differences between sample means of
  Group 1 and Group 2

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Independent t Test

• Independent t Test

• Paired-Sample

• Single observation from each participant from
  two independent groups
• The observation from the second group is
  independent from the first since they come from
  different subjects.
• Comparing a the difference between two means to
  a distribution of differences between mean scores
  .

• Two observations from each participant
• The second observation is dependent upon the
  first since they come from the same person.
• Comparing a mean difference to a distribution of
  mean difference scores

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Independent t Test Steps
Step 1
Step 2
Step 3
Step 4
Step 6
Step 5
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Step 6
Step 7
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Independent t Test

• Similar to previous steps except it takes more
  time to calculate the estimate of the standard
  error, called the pooled estimate of the standard
  error.
• Must calculate Pooled Variance, a weighted
  average of the estimates of the variance from
  both samples.

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