Determine whether H0 should be rejected

Question

Suppose we have a multinomial population with four categories:A, B, C, and D. The null hypothesis is that the proportion of items is the same in every category. A sample size of 300 yields the following results. A=85 B=95 C=50 D=70 Use 0.05 to determine whether H0 should be rejected.

Answer 1

The simplest way is to apply a χ² test. For testing with K categories, the appropriate distribution is χ²(df = K-1) so here we will use χ²(df = 3).

 

The statistic is

 

χ² = ∑ (observed - expected)²/expected

 

= (85 - 75)²/75 + (95 - 75)²/75 + (50 - 75)²/75 + (70 - 75)²/75

 

= (100 + 400 + 625 + 25)/75 = 1150/75 =  15.333...

 

Under the distribution χ²(df = 3), the p-value is

 

P(χ² ≥ 15.333...) = 0.001553...

 

This is less than α = 0.05 so you will reject H₀.