Question

A local brewery produces three premium lagers named Half Pint, XXX, and Dark Night. Of its premium lagers, they bottle 40% Half Pint, 40% XXX, and 20% Dark Night lagers. In a marketing test of a sample of consumers, 29 preferred the Half Pint lager, 40 preferred the XXX lager, and 11 preferred the Dark Night lager. Using a chi-square goodness-of-fit test, decide to retain or reject the null hypothesis that production of the premium lagers matches these consumer preferences using a 0.05 level of significance.

State the value of the test statistic. (Round your answer to two decimal places.)
=

Answer 1

Test Statistic = χ² = 3.84

Solution:

Here, we have to use chi square test for goodness of fit.

Null hypothesis: H₀: Data follows the given distribution.

Alternative hypothesis: H_a: Data do not follow the given distribution.

We assume/given level of significance = α = 0.05

We are given

Number of categories = N = 3

Degrees of freedom = df = N - 1 = 2

α = 0.05

Critical value = 5.991465

(by using Chi square table or excel)

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

Calculation tables for test statistic are given as below:

Category	Prop.	O	E	(O - E)^2/E
Half Pint	0.4	29	32	0.28125
XXX	0.4	40	32	2
Dark Night	0.2	11	16	1.5625
Total	1	80	80	3.84375

Test Statistic = Chi square = ∑[(O – E)^2/E] = 3.84375

χ² = 3.84375

P-value = 0.146332332

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that production of the premium lagers matches these consumer preferences using a 0.05 level of significance.