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t Tests
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Hypothesis Testing

The most basic and commonly used procedures

One Sample t Test
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Two Sample t Test

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Compare a single mean to a fixed number
Compare two population means based on independent samples from
the two populations or groups
Paired t Test

Compare two means based on samples that are paired in some way
One Sample t Test or z test

Compare sample results with a known value
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Mean weight of male Geography majors vs mean weight male
UA students
The advertised tension strength of garbage bags vs your
sample
SAT scores for a random sample vs the mean SAT value
from 3 years ago
Government specification on the percentage of fruit juice
that must be in a drink before it can be sold as fruit juice
Considerations for 1 sample t Test

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If the sample size is small (< 15), t test should
only be used if there is minimal skewness and
outlier impact
If the sample size is moderate (15 – 40), t test
can be used in most cases unless there are
extreme outliers
If the sample size is large (> 40), t test may be
used safely
2 tailed or 1 tailed test

1 tailed

If you are interested in rejecting the null hypothesis if the
population mean differs from the hypothesized value in a
direction of interest

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2 tailed
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Ho: µ = µo (the pop mean is = to the hypothesized value
Ha: µ > µo (the pop mean is > µo)
Ho: µ = µo (the pop mean is = to the hypothesized value
Ha: µ ≠ µo (the pop mean is ≠ to µo)
SPSS reports 2 tailed p value

Divide by 2 for 1 tailed test
2 Sample t Test

Determine whether the unknown means of 2
populations are different from each other based
on independent samples from each population
Equal Variance
Unequal Variance
2 Sample Considerations

Comparison of means

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Data should not be categorical even if it is recoded
Independent Samples
2 samples from same population
 2 samples from different populations

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Check for normality with smaller samples
Equal variances

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Levene’s test
Sample sizes from both groups are similar
paired t Test
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Before and after
Two measurements on the same subject
Control group vs treatment group
Ho: µd = 0 (population meqan of the differences is 0)
 Ha: µd ≠ 0 (population mean of the difference is not
zero)
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Data analyzed are differences within pairs
Paired t Test considerations


Pairing observations may increase the ability to
detect differences
Normality of difference scores
One Sample Test Practice
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Calculate Test Statistic
Calculation of the test statistic requires four components:
The average of the sample (observed average)
The population average or other known value (expected average)
The standard deviation of the average
The number of observations.
T stat- compare to table
Having calculated the t-statistic, compare the t-value with a
standard table of t-values to determine whether the t-statistic
reaches the threshold of statistical significance.
Practice

SE = s / sqrt( n )
t = (x - μ) / SE
We are testing to see if a sample mean score of
freshman GY101 students is less than the class
mean of 70 on exam 1.
65
67
52
58
46
85
44
41
43
50
83
53
49
52
74
39
72
76
Freshman test scores
your list of numbers: 1, 3, 4, 6, 9, 19
SD calculation example
mean: (1+3+4+6+9+19) / 6 = 42 / 6 = 7
list of deviations: -6, -4, -3, -1, 2, 12
squares of deviations: 36, 16, 9, 1, 4, 144
sum of deviations: 36+16+9+1+4+144 = 210
divided by one less than the number of items in the list: 210 / 5 = 42
square root of this number: square root (42) = about 6.48
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