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Hypothesis Testing with t: A Review, Slides of Statistics

An introduction to hypothesis testing using the t-distribution. It covers the steps involved in hypothesis testing, including calculating the appropriate test statistic and critical value, evaluating sample data, and reaching a conclusion. The document also discusses effect size as a way to quantify the magnitude of a treatment effect.

Typology: Slides

2011/2012

Uploaded on 11/14/2012

dharm
dharm 🇮🇳

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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
1. State H0, H1, and choose
α
2. Determine what type of observation it
would take to reject H0
a) What is the appropriate test s tatistic?
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 H0
a) What is the appropriate test s tatistic?
b) What is the critical value?
With a known
µ
, and
σ
, we can calculate
probabilities with Z
Z = M - µ
σM
σ M = σ
n
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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

  1. 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

  1. 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? (calculate df , lookup t )
  3. Evaluate the sample data a) Calculate sample M and s b) Calculate the standard error c) Calculate t
  4. 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 =.

  1. 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