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How to test alternative hypothesis claim?
The heights (in inches) of 20 randomly selected adult males are listed below. Test the claim that the variance is less than 6.25. Use α = 0.05. Assume the population is normally distributed.
70 72 71 70 69 73 69 68 70 71
67 71 70 74 69 68 71 71 71 72
I punched in the numbers in the TI-83 and got the mean, standard deviation. How do I test the claim?
Hypothesis Test for population variance
If we have a sample from an underlying normal distribution and variance σ² then we can test the null hypothesis:
H0: σ² = σ0²
for some fixed σ0².
If H0 is true then Χ² = (n - 1) S² / σ0². Where Χ² is the chi square with n - 1 degrees of freedom.
for the alternate hypothesis we have:
H1a: σ² > σ0²
H1b: σ² < σ0²
H1c: σ² ≠ σ0²
the test statistic is the same for all tests.
the rejection regions for the above tests are:
a) Χ² > Χ²α
b) Χ² < Χ²1-α
c) Χ² < Χ²α/2 or Χ² > Χ²1-α/2
where Χ²α is the value such that:
P(Χ² > Χ²α) = α where Χ² is the chi square with n - 1 degrees of freedom.
In this question we have:
H0: σ² ≥ 6.25 vs. H1: σ² < 6.25
the variance of the sample is: 2.976316
the test statistic is:
(20 - 1) * 2.976316 / 6.25 = 9.048
the p-value is: P(Χ² < 9.048) = 0.02732636
with the low p-value we reject the null and conclude the variance is less than 6.25
Real Estate Test Listing 45