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Introduction to t
I. Hypothesis testing with Z, a review
II. Estimating pop. Standard Deviation
III. The t distribution & table
IV. The t formula
V. Steps
VI. Examples
VII. Effect Size
Hypothesis testing with Z, a review
- State H 0 , H 1 , and choose α
2. Determine what type of observation it
would take to reject H 0
a) What is the appropriate test statistic? b) What is the critical value?
3. Evaluate the sample data
4. Reach a conclusion
Hypothesis testing with Z, a review
2. Determine what type of observation it
would take to reject H 0
a) What is the appropriate test statistic? b) What is the critical value? With a known μ, and σ , we can calculate
probabilities with Z
Z = M - μ σ M σ (^) M =
√ n
Estimating pop. Standard
Deviation
But what if do not know σ , the population
standard deviation? We have to estimate it from our sample: Remember: SS = x- M )^2 Population Sample Variance: σ 2=SS/N s^2 =SS/n- Standard Deviation: σ =^ S/N s= S/n-
Estimating pop. Standard
Deviation
Similarly, estimate the standard error: Also, remember your estimates of population values get better with sample size. Also, df = n-
σ M =
√ n
s M =
s
√ n
s^2
n
The t distribution & table
Previously we argued that the distribution of sample means was normal (central limits theorem), so we could use Z to test hypotheses about those means. The t distribution is a probability distribution (similar to Z) that takes into account that we are estimating the standard deviation, and this estimate changes with degrees of freedom (sample size). The distribution is normal when n is large, but flatter than normal for small values of n.
Steps
- State H 0 , H 1 , and choose α
- Determine what type of observation it would take to reject H 0 a) What is the appropriate test statistic? b) What is the critical value? (calculate df , lookup t )
- Evaluate the sample data a) Calculate sample M and s b) Calculate the standard error c) Calculate t
- Reach a conclusion Examples Are basketball players taller than average? Average males in the US at 5’ 9 ’’ (69”) tall. Here is a random sample of basketball players: 6 ’ 4 ”, 6’ 6 ”, 6’ 2 ”, 6’ 4 ” With alpha equal to .05, test if basketball players as significantly taller than average. Steps
1) H 0 : μ 1 = μ 2
H 1 : μ 1 < μ 2
α =.
- a) use a t (you don’t know σ)
b) t crit (3) > 2.
3) Calculate M, SS, s, sM, t
4) tobs > tcrit, reject H 0
Other examples
These will be done in class as
appropriate.
Effect Size
What if we find a statistically significant
effect, is the effect “meaningful.”
Effect size is a way to quantify the
magnitude of a treatment effect.
Two measures:
Cohen’s d
r^2
Effect Size Cohen’s d Measures how big the effect is by comparing it to the standard deviation. Cohen’s d = mean difference/standard dev. In our basketball example: Cohen’s d = 76-69/1.63 = 4.
Effect Size
Basketball example:
r^2 = t^2 /( t^2 + df) = 8.54^2 /(8.54^2 +3) =. That is, 96% of the variance from the population mean can be explained by the fact that my sample is basketball players. Effect Size
Interpreting r^2
