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T- Statistic used to test hypotheses about an unknown population mean, μ …
T- Statistic
used to test hypotheses about an unknown population mean, μ
,when the value of s is unknown. The formula for the t statistic has the same structure as the z-score formula, except that the t statistic uses the estimated standard error in the denominator.
Estimated Standard Error (sM)
Definition: Estimate of standard distance between M and μ
sM = s / √n.
s = sample standard deviation
n = sample size
Example.
If s = 6.02 and n = 20:
sM = 6.02 / √20 = 6.02 / 4.47 = 1.35
T Statistic.
Hypothesis testing tool using estimated standard error
t = (M - μ) / sM
Example-M = 24.4, μ = 27.2, sM = 1.35:
t = (24.4 - 27.2) / 1.35 = -2.07
M = sample mean
μ = hypothesized population mean
sM = estimated standard error
The Problem with Z-Scores
z = (M - μ) / σM
where σM = σ / √n
Z-score formula requires: σ (population standard deviation)
Reality: σ is usually unknown
Paradox: Need to know population to test population
Use Sample Values as Estimates
Replace unknown population parameters with sample statistics
Population (Unknown)
σ = ?
Sample (Known)
s = sample standard deviation
Degrees of Freedom (df)
Number of scores in a sample that are independent and free to vary
df = n - 1We lose 1 df because the sample mean restricts one value
Example 1 -
Sample size n = 25
df = 25 - 1 = 24
Example 2-
Sample size n = 40
df = 40 - 1 = 39
T Distribution-
Complete set of t values for every possible random sample of a given size
Shape
Bell-shaped, symmetric
Heavier tails than normal distribution
Mean
0
Variation
Shape depends on df
Approaches normal as df increases
Estimated Cohen's d-
Measures effect size (standardized mean difference)
d = (M - μ) / s
Small: d = 0.2
Medium: d = 0.5
Large: d = 0.8
Example-M = 8.1, μ = 7.4, s = 0.8:
d = (8.1 - 7.4) / 0.8 = 0.88
Large effect!
Confidence Interval-
Range of values centered around sample statistic that likely contains population parameter
CI = M ± tcritical × sM-
Common confidence levels: 80%, 85%, 90%, 95%, 99%
Example-(95% CI):
M = 36.7, tcritical = 1.97, sM = 1.34:
Margin of error = 1.97 × 1.34 = 2.64
CI = 36.7 ± 2.64 = [34.1, 39.3]
We are 95% confident μ is between 34.1 and 39.3
T-Test vs Z-Test: When to Use
Use T-Test When
Population σ is unknown
✓ Small sample size (n < 30)
✓ Using sample standard deviation (s)
✓ Most real-world research situations
Example- Testing a new literacy program with 20 students - population data doesn't exist yet!
Use Z-Test When Population σ is known
✓ Large sample size (n ≥ 30)
✓ Population is normally distributed
✓ Rare in practice
Example-
Testing if your class differs from national standardized test (where σ is published)