The t-distribution

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The t-distribution
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General comment on z and t
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Moving from z to t

• Same concept, different assumptions
• Can only use z-tests if you know population SD
• So when you have to estimate s, use t-dist.
• t-test estimates population SD from sample SD
• t-test more robust against departures from
  normality (doesnt affect the accuracy of the
  p-estimate as much)

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Calculating the t-statistic

• We dont know anything about the population

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Step 1 Estimate s
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Calculating the t-statistic

• We dont know anything about the population

?
Step 2 Use to estimate SEM
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Calculating the t-statistic

• We dont know anything about the population

?
Step 3 Use these in t-statistic
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t-statistic factors in significance

• Size of estimated SE obviously depends on both SD
  of sample, and sample size
• Thus, factors affecting size of calculated t are
  mean diff, sample SD, and sample size

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Sampling distribution of t

• The t ratio requires 2 sample statistics to be
  used to estimate population parameters (sample
  mean and sample standard error)
• The Z-ratio only required one (sample mean)

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Sampling distribution of t

• So, sampling variation in Z-distribution
  reflected variability with respect to sample mean
• BUT sampling variation in t-distribution reflects
  variability with respect to sample mean and
  standard error of the mean
• Soas the sample gets smaller (and the standard
  error of the mean then increases) wed expect the
  sampling distribution of t to differ from that of
  Z
• The good old 1.96 for 95 is toast

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Sampling distribution of t
Large n ? t-dist pretty much like the
z-dist (because sample SD is a good estimate of
pop SD, sample SE is a good estimate of pop SE)
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Sampling distribution of t
Small n ? t-dist departs from the z-dist (because
sample SD is a poor estimate of pop SD, sample
SE is a poor estimate of pop SE)
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Sampling distribution of t
a (Significance level)
df n-1
Because distribution gets flatter as n gets
smaller, this implies t for significance gets
bigger as n gets smaller
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(digression degrees of freedom)

• Degrees of freedom
• The number of independent pieces of information a
  sample of observations can provide for purposes
  of statistical inference
• E.g. 3 numbers in a sample 2, 2, 5
• Sample mean 3 deviations are 1, -1, 2
• Are these independent?
• No when you know two, youll know the other
  because of
• In other words, for any sample of size n you
  have n-1 thing that are free to vary the
  other one is fixed

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Confidence intervals for t
Standard error of the mean

• General rule

Sample mean
t-statistic (changes with different df and ?)
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Independent t-tests

• Uses a sampling distribution of differences
  between means

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Lets think

• Comparing 2 means from the same population
• If we sampled enough, what would we expect the
  mean difference to be?
• What would influence the accuracy of this
  expectation?

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Comparing 2 means

• Estimating the mean of the sampling distribution
  of differences in the means
• Estimating the standard error of the differences
  in the means
• Getting a little complicated

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Comparing 2 means

• Estimating the standard error of the differences
  in the means
• Must assume equal population variance in the two
  samples
• So this is an assumption of independent t-tests
  that must be tested
• Then
• Know that

the SD of the distribution of differences between
2 sample means
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Comparing 2 means
Difference between sample means

• T-test for 2 independent samples

Note
SEM of difference between sample means
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Recall

• Larger sample size, and less variability in
  population imply...
• ...reduced variability in the distribution of
  sampling means

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Extending to 2 sample means

• With larger samples, it is less likely that the
  observed difference in sample means is
  attributable to random sampling error
• With reduced variability among the cases in each
  sample, it is less likely that the observed
  difference in sample means is attributable to
  random sampling error
• With larger observed difference between two
  sample means, it is less likely that the observed
  difference in sample means is attributable to
  random sampling error
• See applet
• http//physics.ubishops.ca/phy101/lectures/Beaver/
  twoSampleTTest.html

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Evaluating tobserved for 2 samples

• The d of f changes from the one-sample case
• comparing two independent means

becomes
If the 2 groups are of equal size
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Reporting t-test in text
Descriptive statistics for the time to exhaustion
for the two diet groups are presented in Table 1
and graphically in Figure 1. A t-test for
independent samples indicated that the 44.2 (?
2.9) minute time to exhaustion for the CHO group
was significantly longer than the 38.9 (? 3.5)
minutes for the regular diet group (t18 - 3.68,
p ? 0.05). This represents a 1.1 increase in
time to exhaustion with the CHO supplementation
diet.
In discussion, address whether the statistically
significant difference is meaningful
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Reporting t-test in table

• Descriptives of time to exhaustion (in minutes)
  for the 2 diets.

Group n Mean SD
Reg Diet 10 38.9 3.54
CHO sup 10 44.2 2.86
Note indicates significant difference, p ? 0.05
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Reporting t-test graphically
Figure 1. Mean time to exhaustion with different
diets.
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Reporting t-test graphically
Figure 1. Mean time to exhaustion with different
diets.
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Summary of theindependent t-test

• Utilize when the assumption of no correlation
  between the groups is valid
• Compares the difference in means by evaluating
  the observed magnitude of the mean difference to
  the expected variability in the magnitude of the
  mean differences when Ho is true.

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t-tests in SPSS

• First note the data format one continuous
  variable (in this case, age)

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t-tests in SPSS

• Second, run the procedure

drag the test variable over
and specify µ
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t-tests in SPSS

• Third, check the output

N, Mean, SD, SEM
significance (if a .05, then lt .05 is
significant)
df n-1 19
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independent-tests in SPSS

• First, check the data

One grouping variable
One test variable
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independent-tests in SPSS

• Second, run the procedure

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independent-tests in SPSS

• Second, run the procedure

1. slide variables over
2. click define groups
3. define groups
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independent-tests in SPSS

• Third, examine the output

N, Mean, SD, SEM
test for equal variances (gt .05 is good)
significance (if a .05, then lt .05 is
significant)