Z & t-tests

• It refers to the statistical tests built on the assumption that the data being analyzed were sampled from a specified distribution

• Parameters of the distribution (e.g. µ & σ2) are estimated and are used to calculate tail probabilities

• Most statistical tests specify the normal distribution

Parametric Analyses

Steps of Parametric Analyses

1. State null and alternative hypotheses

2. Specify the null distribution

3. Obtain sample data

4. Calculate the test statistic & tail probability based on threshold criterion, α

5. Decide to reject or fail to reject the null hypothesis and interpret your decision

Rejecting or Failing to Reject the H0

• If the test statistic falls in the critical region, the null hypothesis is rejected

• If the test statistic does not fall in the critical region, the null hypothesis is NOT rejected

• Notice that no statements are made about the alternative hypothesis

Z-test

• Obtains the probability of sample means being within any interval or beyond any point

• Assumptions: know the population mean and standard deviation and have normally-distributed data

• Thus, we can test simple hypotheses about sample means (called the Z-test) – Calculate z-score for sample mean, zobt

– Compare to critical z-score, zcrit

Who did better on the their test?

Dick Harry

X=70 Y=80

µ=65 µ=70

σ=2 σ=10

Formula for a Z-score

= standard error = standard

deviation of the sampling

distribution

Who did better on the their test?

Dick: (70-65)/(2/(1^0.5)) = 2.5 st. dev.

Harry: (80-70)/(10/(1^0.5)) = 1.0 st. dev.

Use Excel Normsdist function instead

of a unit normal table

• The probability is 0.773 or 1 - 0.227

Example

– Find the z-score associated with the

upper and lower scores when considering

95% of a normal distribution • “upper and lower scores”: two-tailed test

• α should be divided by 2 before looking up the z-

score

• α/2 = 0.05/2 = 0.025

• ± 2 stdev. ≈ 95% of distribution

t- distribution

http://www.statsoft.com/textbook/sttable.html#t

2. Specify the Test Statistic:

t test

21

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XXt

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Where this equals the standard error of the

difference between the two means, and df1=n1-1,

df2=n2-1, and dft=n1+n2-2

Conduct the Correlated t-test

Simulation

• http://onlinestatbook.com/simulations/corre

lated_t/correlated_t.html

Generalized Confidence Intervals

P(Y – tα[n-1]sY ≤ µ ≤ Y + tα[n-1]sY) = (1-α)

• where tα[n-1] is the critical value of the t-

distribution with probability P = α, and sample

size n, and sY is the standard error of the mean

• Calculate 95% confidence interval for

previous example with five difference

scores

• 3.04-6.96

Comparison of Z-test to single

sample t-test H0: µ = 58, HA: µ ≠ 58, α=0.10, n=20, σ2=28, X=56

Z-test

z=(56-58)/((28/20)^0.5) = -1.69 reject H0 because -1.69<-1.645 Zcrit

90% confidence interval = 56±1.645*((28/20)^0.5) = 54.054-57.946

t-test

t=(56-58)/((28/20)^0.5) = -1.69 fail to reject H0 because -1.69>-1.729 tcrit for α=0.10 and df=19

90% confidence interval = 56±1.729*((28/20)^0.5) = 53.955-58.045

t-test Equation Summary

• Single sample t = (Y-µH0)/sY

• Dependent sample t-test = (D-µH0)/sD

– where sYorD = sYorD / √n

• Independent sample t-test=((X1-X2)-µH0)/sX1-X2

– where

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