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Random
Sample
Statistics
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Chi-square (χ2) Distribution
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t Distribution
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F Distribution
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•As n gets large the bias goes to zero
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Null
Alternative Hypothesis
Hypothesis
•In this example, H1 is a two-sided alternative hypothesis
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•H1 in equation 3-22 is a two-sided alternative hypothesis
•The procedure for testing this hypothesis is to:
− take a random sample of n observations on the random variable x,
− compute the test statistic, and
− reject H0 if |Z0| > Zα/2, where Zα/2 is the upper α/2 percentage of the
standard normal distribution.
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One-Sided Alternative Hypotheses
• In some situations we may wish to reject H0
only if the true mean is larger than µ0
– Thus, the one-sided alternative hypothesis is H1:
µ>µ0, and we would reject H0: µ=µ0 only if Z0>Zα
• If rejection is desired only when µ<µ0
– Then the alternative hypothesis is H1: µ<µ0, and
we reject H0 only if Z0<−Zα
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Confidence Intervals
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Confidence Interval on Mean, Variance Known
Furthermore, a 100(1 − α)% upper confidence bound on µ is
whereas a 100(1 − α)% lower confidence bound on µ is
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• For the two-sided alternative hypothesis, reject H0 if |t0| > tα/2,n-1, where
tα/2,n-1, is the upper α/2 percentage of the t distribution with n − 1 degrees of
freedom
• For the one-sided alternative hypotheses,
• If H1: µ1 > µ0, reject H0 if t0 > tα,n − 1, and
• If H1: µ1 < µ0, reject H0 if t0 < −tα,n − 1
• One could also compute the P-value for a t-test
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• Section 3-3.4 describes hypothesis testing and confidence intervals
on the variance of a normal distribution
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• Section 3-4.3 describes hypothesis testing and confidence intervals
on the variance of two normal distributions
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