Psychology 315: Statistics I

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Introduction to the t-test
Aron, Aron Coups, Chapter 8
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The t-distribution

• We have made use of the z distribution (the
  standard normal distribution) to do hypothesis
  testing and to compute confidence intervals for
  example
• For hypothesis testing we can calculate
• Z (M - PopulationM ) / PopulationSDM
• and determine if Z exceeds our z-score cutoff
  (e.g., 1.64)
• For confidence intervals we can calculate
• Confidence interval M 1.96(PopulationSDM)
• These two calculations require knowing
  PopulationSD so that we can calculate
  PopulationSDM
• What if we didnt know PopulationSDM and had to
  estimate it from our sample?
• What should we use to estimate PopulationSDM ?
• Would we be able to substitute this estimate
  (estimated PopulationSDM) into our formulas and
  have everything work out the same way?

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Sampling Distributions -- biased and unbiased
estimates

• We know that if you repeatedly choose samples of
  size N and compute the mean (M) for each sample,
  we will create a distribution of sample means.
• The mean of the distribution of sample means
  (PopulationMM) will equal that of the original
  distribution (PopulationM).
• In this sense the sample mean (M) is an unbiased
  estimate of PopulationM because on average M
  PopulationM.

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Sampling Distributions -- biased and unbiased
estimates

• If we repeatedly choose samples of size N and
  compute the variance (SD2) for each sample,
• we will create a distribution of variances.
• The mean of the distribution of variances will
  not equal of the variances of original
  distribution (PopulationSD2).
• In this sense the sample variance (SD2) is a
  biased estimate of PopulationSD2 because on
  average SD2 ? PopulationSD2.

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Sampling Distributions -- biased and unbiased
estimates

• But, if we repeatedly choose samples of size N
  and compute the the variance as
• we will create a distribution of variances and
  the mean of the distribution of variances
  (computed with N-1) will equal of the variance of
  original distribution (PopulationSD2).
• We refer to N-1 as the number of degrees of
  freedom (df) i.e., df N-1
• In this sense the sample variance computed with
  N-1 in the denominator (S2) is an unbiased
  estimate of PopulationSD2 because on average S2
  PopulationSD2.

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Sampling Distributions -- biased and unbiased
estimates

• Therefore, going back to our first question,
• What should we use to estimate PopulationSDM ?
• The answer is we should first compute an unbiased
  estimate of PopulationSD2 as
• And then we should compute an estimate of
  PopulationSDM as
• or
• Explain the difference between S2 and SD2
• Explain the difference between S and SM
• Explain the difference between SM and
  PopulationSDM

Web Demo
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The t-distribution

• What we know already
• The distribution of sample means has the
  following parameters
• PopulationMM PopulationM
• PopulationSDM PopulationSD/sqrt(N)
• Thus any particular M can can be converted to a Z
  score in the following way
• Producing the standard normal distribution or,
  in other words, a sampling distribution of
  Z-scores.

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The t-distribution

• Gosset raised the following question
• How would things change if we divided by SM
  rather than PopulationSDM itself?
• Would the resulting sampling distribution be
  normal?
• The answer is no in general t-scores are not
  normally distributed.
• But the distribution of t-scores is symmetric
• and has a mean of zero
• The exact form of the t-distribution depends on N
• For large N the t-distribution approaches the
  standard normal distribution (the z-distribution)
  but for small N the t- distribution becomes
  leptokurtic.

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The t-distribution
The good news is that the t-distribution has a
definite form we know this because of smart
people like Gossett
You dont have to remember this!!!
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The t-distribution
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The t-distribution
We use the t-distribution in same way we use the
z-distribution we find t-critical in the same
way we find z-critical
t-critical (tcrit)
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The one sample t-test

• Consider our baby/supercharged vitamin example
  again
• Lets say you know that on average the first
  steps are taken at 14 months
• but you dont know PopulationSD

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The one sample t-test

• Assume youve chosen a sample of 16 babies and
  given them the vitamins
• H0 Pop1M gt Pop2M
• H1 Pop1M lt Pop2M
• Set ? .05 and perform a one tailed t-test.
• The comparison distribution is the t-distribution
  
• with df (N - 1) 15, and tcrit -1.753
• Calculate t

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The one sample t-test

• The sample mean is M 8 and and standard
  deviation S 12
• Calculate SM S / sqrt(N) 12/4 3
• Calculate t (M - PopulationM) / SM (8 - 14) /
  3 -2
• t -2, which is less than tcrit -1.753
• Therefore, reject H0.
• Conclude that the vitamins had an effect.

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The one sample t-test

• Consider our rat on the running wheel again
• Lets say you know that on average rats spend 50
  minutes out of 2 hrs on a running wheel but you
  dont know PopulationSD

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The one sample t-test

• Assume youve chosen a sample of 25 rats and
  injected them with an amphetamine then measured
  the time they spend on the running wheel.
• H0 Pop1M ??? Pop2M
• H1 Pop1M gt Pop2M
• Set ? .05 and perform a one tailed test.
• The comparison distribution is the t-distribution
  with df 24, therefore tcrit 1.711
• Calculate t

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The t-test for dependent means

• The sample mean is M 60 and and standard
  deviation S 16
• Calculate SM S / sqrt(N) 16/5 3.2
• Calculate t (M - PopulationM) / SM (60 - 50)
  / 3.2 3.125
• t 3.125 which is greater than tcrit 1.711
• Therefore, reject H0.
• Conclude that the amphetamine injections had an
  effect.

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The t-test for dependent means

• Consider our rats on the running wheel again
• Repeated measures designs.

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The t-test for dependent means

• Assume youve chosen a sample of 10 rats and
  measured their running time before and after an
  amphetamine injection.
• Before After Difference
• 12 13 1
• 14 14 0
• 15 16 1
• 13 13 0
• 12 13 1
• 11 13 2
• 14 14 0
• 13 13 0
• 0.625 mean Mdiff
• 0.744 S
• 0.263 SM S / sqrt(N)
• 2.38 t (t Mdiff / SM)

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The t-test for dependent means

• Our t score 2.38, which exceeds the cutoff
  value of t (1.89) for a one tailed test.
• Therefore we reject the null hypothesis.

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The t-test for dependent means

• Assumption the t-test is based on the assumption
  that the population you sampled is normal
• If this assumption does not hold then the
  validity of the t-test may be questions
• One way to assess this assumption is to see
  whether the sample distribution appears to be
  normal.
• Estimating Effect Size
• Because we dont know the population SD we cant
  compute effect size but we can estimate it from
  the mean difference and the sample standard
  deviation
• i.e., M / S

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The t-test for dependent means

• Power

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The t-test for dependent means

• Power when Planning an Experiment

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Confidence intervals for means and mean
differences

• Recall that when we knew the PopulationSD and
  sample size, we were able to compute a confidence
  interval about a sample mean
• i.e., M zcrit (PopulationSDM)
• When we have to estimate PopulationSD from S, we
  can use SM and the t-distribution to calculate a
  condidence interval about a sample mean
• i.e., M tcrit (SM)
• The same procedure can be used to place a
  confidence interval around a mean difference (or
  a mean) when PopulationSD is unknown
• i.e., MDiff tcrit (SM) where SM is the standard
  error of the difference scores.