Testing means, part II The paired t-test

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Testing means, part IIThe paired t-test
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Outline of lecture

• Options in statistics
• sometimes there is more than one option
• One-sample t-test review
• testing the sample mean
• The paired t-test
• testing the mean difference

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A digressionOptions in statistics
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Example

• A student wants to check the fairness of the
  loonie
• She flips the coin 1,000,000 times, and gets
  heads 501,823 times.
• Is this a fair coin?

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Ho The coin is fair (pheads 0.5).
Ha The coin is not fair (pheads ? 0.5).
n 1,000,000 trials x 501,823 successes
Under the null hypothesis, the number of
successes should follow a binomial distribution
with n1,000,000 and p0.5
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Test statistic
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Binomial test

• P 2PrX501,823
• P 2(PrX 501,823 PrX 501,824 PrX
  501,825 PrX 501,826
• ...
• PrX 999,999 PrX 1,000,000

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Central limit theorem
The sum or mean of a large number of
measurements randomly sampled from any population
is approximately normally distributed
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Binomial Distribution
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Normal approximation to the binomial distribution
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Example

• A student wants to check the fairness of the
  loonie
• She flips the coin 1,000,000 times, and gets
  heads 501,823 times.
• Is this a fair coin?

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Normal approximation

• Under the null hypothesis, data are approximately
  normally distributed
• Mean np 1,000,000 0.5 500,000
• Standard deviation
• s 500

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Normal distributions

• Any normal distribution can be converted to a
  standard normal distribution, by

Z-score
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From standard normal table P 0.0001
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Conclusion

• P 0.0001, so we reject the null hypothesis
• This is much easier than the binomial test
• Can use as long as p is not close to 0 or 1 and n
  is large

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Example

• A student wants to check the fairness of the
  loonie
• She flips the coin 1,000,000 times, and gets
  heads 500,823 times.
• Is this a fair coin?

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A Third Option!

• Chi-squared goodness of fit test
• Null expectation equal number of successes and
  failures
• Compare to chi-squared distribution with 1 d.f.

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Test statistic 13.3 Critical value 3.84
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Coin toss example
Chi-squared goodness of fit test
Normal approximation
Binomial test
Most accurate Hard to calculate Assumes Random
sample
Approximate Easier to calculate Assumes Random
sample Large n p far from 0, 1
Approximate Easier to calculate Assumes Random
sample No expected lt1 Not more than 20 less
than 5
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Coin toss example
Chi-squared goodness of fit test
Normal approximation
Binomial test
in this case, n very large (1,000,000) all P lt
0.05, reject null hypothesis
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Normal distributions

• Any normal distribution can be converted to a
  standard normal distribution, by

Z-score
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t distribution

• We carry out a similar transformation on the
  sample mean

mean under Ho
estimated standard error
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How do we use this?

• t has a Student's t distribution
• Find confidence limits for the mean
• Carry out one-sample t-test

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t has a Students t distribution
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t has a Students t distribution
Uncertainty makes the null distribution FATTER
Under the null hypothesis
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Confidence interval for a mean
?(2) 2-tailed significance level df degrees
of freedom, n-1 SEY standard error of the mean
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Confidence interval for a mean
95 Confidence interval Use a(2) 0.05
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Confidence interval for a mean
c Confidence interval Use a(2) 1-c/100
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One-sample t-test
Null hypothesis The population mean is equal to
?o
Sample
Null distribution t with n-1 df
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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The following are equivalent

• Test statistic gt critical value
• P lt alpha
• Reject the null hypothesis
• Statistically significant

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Quick reference summary One-sample t-test

• What is it for? Compares the mean of a numerical
  variable to a hypothesized value, µo
• What does it assume? Individuals are randomly
  sampled from a population that is normally
  distributed
• Test statistic t
• Distribution under Ho t-distribution with n-1
  degrees of freedom
• FormulaeY sample mean, s sample standard
  deviation

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Comparing means

• Goal to compare the mean of a numerical variable
  for different groups.
• Tests one categorical vs. one numerical variable

Example gender (M, F) vs. height
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Paired vs. 2 sample comparisons
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Paired designs

• Data from the two groups are paired
• There is a one-to-one correspondence between the
  individuals in the two groups

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More on pairs

• Each member of the pair shares much in common
  with the other, except for the tested categorical
  variable
• Example identical twins raised in different
  environments
• Can use the same individual at different points
  in time
• Example before, after medical treatment

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Paired design Examples

• Same river, upstream and downstream of a power
  plant
• Tattoos on both arms how to get them off?
  Compare lasers to dermabrasion

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Paired comparisons - setup

• We have many pairs
• In each pair, there is one member that has one
  treatment and another who has another treatment
• Treatment can mean group

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Paired comparisons

• To compare two groups, we use the mean of the
  difference between the two members of each pair

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Example National No Smoking Day

• Data compares injuries at work on National No
  Smoking Day (in Britain) to the same day the week
  before
• Each data point is a year

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data
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Calculate differences
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Paired t test

• Compares the mean of the differences to a value
  given in the null hypothesis
• For each pair, calculate the difference.
• The paired t-test is a one-sample t-test on the
  differences.

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Hypotheses
Ho Work related injuries do not change during No
Smoking Days (µ0) Ha Work related injuries
change during No Smoking Days (µ?0)
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Calculate differences
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Calculate t using ds
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Caution!

• The number of data points in a paired t test is
  the number of pairs. -- Not the number of
  individuals
• Degrees of freedom Number of pairs - 1

Here, df 10-1 9
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Critical value of t
Test statistic t 2.45
So we can reject the null hypothesis Stopping
smoking increases job-related accidents in the
short term.
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Assumptions of paired t test

• Pairs are chosen at random
• The differences have a normal distribution
• It does not assume that the individual values are
  normally distributed, only the differences.

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Quick reference summary Paired t-test

• What is it for? To test whether the mean
  difference in a population equals a null
  hypothesized value, µdo
• What does it assume? Pairs are randomly sampled
  from a population. The differences are normally
  distributed
• Test statistic t
• Distribution under Ho t-distribution with n-1
  degrees of freedom, where n is the number of
  pairs
• Formula