z & t-tests - facstaff home page for...
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Z & t-tests
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• 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
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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
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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
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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
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Who did better on the their test?
Dick Harry
X=70 Y=80
µ=65 µ=70
σ=2 σ=10
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Formula for a Z-score
= standard error = standard
deviation of the sampling
distribution
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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.
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Use Excel Normsdist function instead
of a unit normal table
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• The probability is 0.773 or 1 - 0.227
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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
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t- distribution
http://www.statsoft.com/textbook/sttable.html#t
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2. Specify the Test Statistic:
t test
21
21
XXs
XXt
21
222
211 11**
21 NNdf
sdfsdfs
TXX
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
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Conduct the Correlated t-test
Simulation
• http://onlinestatbook.com/simulations/corre
lated_t/correlated_t.html
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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
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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
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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
21
222
211 11**
21 NNdf
sdfsdfs
TXX
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