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Chapter 9: The t Statistic - Coggle Diagram
Chapter 9: The t Statistic
t-Statistic: A statistic similar to a z-score that can be used to test hypotheses about an unknown population mean when the population standard deviation is not known
Has the same formula as the z formula, but uses estimated standard error in place of standard deviation
Estimated Standard Error: Equal to the sample standard deviation divided by the root of the sample size, this estimates the variance in the population
Remember that estimated standard error is a product of sample variance
Is useful because usually population standard deviation is not a known quantity
Degrees of Freedom: Number of scores in a sample that are free to vary. For a sample, this is equal to n-1 if a sample mean is known
t Distribution: Complete set of values computed for every possible random sample for a specific sample size or specific degrees of freedom. Similar to a distribution of sample means
Since the distribution of sample means is normal at n>=30, so will the t distribution be
Tend to be more flat and spread out than z distributions
This is because the sample variance changes from one sample to another
t Table: Similar to the unit normal table, the t table is a list of all proportions for all values of t
Unlike the unit normal table, degrees of freedom appears in one column as degrees of freedom will change the t value
If your t table is abridged, use the next
smallest
df value, not the
closest
value
This is to prevent Type I errors
Hypothesis Testing with the t Statistic
t Test Value: Equals the sample mean minus the population mean (from the null hypothesis), divided by the estimated standard error (from the sample data)
The value from the null hypothesis can be based in logic, theory, or simply wishful thinking, though ideally it will be established from a control group
Keep an eye on how researchers are creating their t values!
Being able to use other values does enable researchers to perform a t test even without a control group, though
Necessary Assumptions
Independent Observations
Sampled population must be normal
This becomes less relevant at larger sample sizes though (n>=30 again)
Measuring Effect Size
Estimated Cohen's d: Equal to the difference in means (population and sample) divided by the sample standard deviation
Reporting Cohen's d represents effect size, which can be helpful as t tests are not great at showing change if the effect size is small
Evaluating Effect by Cohen's d
d=.2: Small effect
d=.5: Medium effect
d=.8: Large effect
Percentage of Variance (r squared): A proportion that compares the variance accounted for by the treatment effect with the total variance in the population
Sometimes represented by lowercase omega squared when dealing with t statistics
Can be calculated two ways
Subtracting the treatment effect from each score, finding the sum of squares for both the scores with and without the treatment effect removed, then dividing the ss from the treatment removed scores by the ss of the treatment included scores
Dividing the square of the t statistic by the square of the t statistic plus the degrees of freedom
Evaluating Effect by r^2
r^2=0.01: Small effect
r^2=0.09: Medium effect
r^2=0.25: Large effect
Confidence Interval: A range of values centered around a sample statistic that estimates the population parameter is somewhere within that range of the sample statistic
Equation: Population mean equals sample mean plus or minus t times the standard error
Increasing the confidence widens the range, as it is more likely that the true value will be within the interval. Decreasing does the opposite
As usual, increasing sample size narrows the confidence interval because you can be more sure you are being accurate
Because confidence intervals are affected by sample size, they do not provide the same absolute measure of effect as Cohen's d
Reminders from previous chapters!
A smaller sample size means more variance means more error, means a smaller t value
A larger sample size means less variance, means less error, means a larger t value
In the case of a one-direction (one-tailed) test, the critical region only concerns the direction of the test