Introduction to the t test
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Transcript of Introduction to the t test
t-Tests give us more options
Inferential
Statistics
What we know already:
Frequency distributions shown in graphs
Summarizing data sets
Central tendency (mean, median, mode)
Variability (variance and standard deviation)
Central Limit Theorem
Sample means are normally distributed
If n > 30 or population is normally distributed
Hypothesis Testing basics
Null hypothesis & critical region defined
Sample data used to make a decision
Bad news:
“The shortcoming of using a z-score as an inferential statistic is that the z-score formula requires more information than is usually available.”
“Specifically, a z-score requires that we know the value of the population standard deviation (or variance). In most situations, however, the standard deviation for the population is not known.”
One-Sample t-Test
We want to compare a single sample to a
population mean, or a hypothetical mean
We don’t know σ2 so we use s2, the Sample Variance
We have an estimated standard error, sM
We use sM to compute t t is similar to z in shape: bell curve
It is shorter in the center, and the tails have more area
There is more than one t curve The curve is determined by M, s2, and df
Degrees of freedom, df is related to sample size
Distribution of the t statistic for different values of degrees of freedom are
compared to a normal z-score distribution. Like the normal distribution, t
distributions are bell-shaped and symmetrical and have a mean of zero. However, t
distributions have more variability, indicated by the flatter and more spread-out
shape. The larger the value of df is, the more closely the t distribution approximates
a normal distribution.
One-Sample t-Test
The same 4 steps of hypothesis testing
State the hypotheses, choose the alpha level
Locate the critical region on the table of t values
Collect sample data and compute t statistic
Evaluate H0 and make a decision
We can measure effect size with Cohen’s d and
with r 2 (proportion of variability explained)
We can have directional (one-tailed) t-tests
Comparison of z- and t-Tests
One-sample z-test
Sample is drawn from a
population whose mean is
known or hypothesized
σ2 is known, compute σM
z=(M-µ) / σM
One sample t-test
Sample is drawn from a
population whose mean is
known or hypothesized
σ2 not known
Use s2 to compute sM
df = n-1
t = (M-µ) / sM
2
Mn
2
2 so M
SSSS s df
s sdf n n
One-sample t-Test
One random sample of interval/ratio variable
Comparison with population
or hypothesized mean
Can be either a
one-tailed or a
two-tailed test
df = n-1
n
s
xt
2
Degrees of Freedom
Always related to the sample size
Symbol: df
Differs across statistical tests
In t-tests, Degrees of Freedom depends on the
number of means that are computed:
One sample mean: df = N – 1
Two sample test: df = N – 2
Repeated measures test computes the mean of the
change score (one mean) so df = N – 1
Sample Problem
A group of telemarketers averaged 80 sales per day.
A sample of n=16 people were randomly chosen for
training on a new technique. After training, the
sample averages 85 sales per day, with SS=60.
Does the sample provide sufficient evidence to
conclude that the training leads to higher levels of
sales?
THINK FIRST
We cannot draw the population distribution,
because we do not know its standard deviation.
We can draw the distribution of sales scores
(the scores of the 16 people) because we know the
mean and standard deviation of the sample.
So: draw the curve representing the sample of
sales scores after training.
What range contains about 95% of scores?
Does it include the pre-training mean of 80?
Hypothesis Test: t-Test
Step 1: Define Hypotheses
H0:_______________________________________
HA: _______________________________________
Step 2: Set Criteria for a decision
Alpha = .05 in two-tails combined
Compute df
Use Table or Calculator to find Critical Value of t
Critical Value is t = __________________________
Define Critical Region (draw yourself a picture)
t-Test Continued
Step 3: Collect data and compute test statistics
Means are given to us: M=85, µ=80
Need to compute s2 in order to compute sM
s2 = _______________
sM = _______________
Compute t = (M-µ) / sM
t = ________________
Step 4: Make a decision based on criteria
Use your picture of the t-curve
Effect Size: Cohen’s d
Effect size has the same meaning for the t-Test that it
did with the z-Test
Equation for Cohen’s d used the population standard
deviation: now we do not have that.
Substitute the sample standard deviation
Compute Cohen’s d for this problem
df
SS
M
ndev' std sample
differencemean d sCohen'
Effect size: r2
Another way of measuring effect size
Out of the total variability, how much is accounted for by being in the treatment group?
SS = Σ(X - M)2 = total variability
Variability without treatment effect
Subtract treatment effect (M-µ) from each score
Then subtract Mean, square the difference, and sum
Compute r2 for this problem
22
2
tr
t df
How to Write Results
1. First sentence: What question was tested? Can
be a statement or a question.
2. Procedure / sample: How was the test done?
Who was tested?
3. What were the results? Report sample statistics
(M and sM), the test statistic (t) and its df,
the significance (p), and decision.
4. If significant, report effect size
5. Close with a summary sentence
Sample Narrative
A sample of 16 telemarketers received special
training. After training, their average daily sales
(M = 85, s = 2) were significantly higher than the
pre-training average of 80 for all the telemarketers
(t(15) = 10, p < .05). The effect size, measured
by Cohen’s d, is quite large (d = 2.5).
Approximately 87% of the variability is accounted
for by the training (r2 = .8695). The sales training
produced a significant and sizable difference,
leading to an increase in sales for the firm.
Types of t - Tests
One sample t-Test
Sample mean compared to hypothetical mean or μ
df = 1 because one sample mean is computed
Independent samples t-Test
Two samples are compared to each other
df = 2 because two sample means are computed
Correlated/Paired/Repeated Measures t-test
Two related measures; mean difference is computed
df = 1 because one mean is computed
Basic dynamic of all of our tests
They will involve a ratio (a fraction)
The numerator (top) will measure variability
between group(s)
The denominator (bottom) will measure the
variability that is due to random chance
“Difference on the top, and error on the bottom”
Statz Rappers