(6) Further hypothesis tests

If a random sample of n observations X₁, X₂, …, X_n is selected from N(µ,σ²), then

(n-1)S²/σ² ~ X²_n-1

100(1-α)% confidence intervals (where α is probability variance falls outside limits)

Hypothesis testing for variance of normal distribution

H_o: σ² = population variance

H₁: σ² ><≠ population variance

Specify sig level

Specific degrees of freedom

Calculate critical region

Identify σ², s², and calculate test statistic 1/σ²(n-1)s²

Conclude

For a random sample of n_x observations from an N(µ_x,σ_x²) distribution and an independent random sample of n_y observations from an N(µ_y,σ_y²) distribution

(Use tables for F test/distribution)

If a random sample of n_x observations is taken from a normal distribution with unknown variance σ² and an independent random sample of n_y observations is taken from normal distribution with equal but unknown variance then

(Use tables for F test/distribution)

F_v1,v2(p)

(F_v2,v1(1-p))^-1

Test whether two variances are the same (F test)

1) Find s²₁ and s²₂, the larger and smaller variances respectively

2) Write down null hypothesis

3) Alternative hypothesis

4) Look up critical value of F_vl,vs in tables (where v_L is degrees of freedom of larger variance and vice versa) (If two-tailed test half sig level e.g. 10% use F_vL,vs(0.05) as critical value)

5) Calculate critical region

6) Calculate F_test = s_L² / s_s²

7) Conclusion