Problem Statement
A university surveyed 200 students regarding their preferred study beverage (Coffee, Tea, Water) and their faculty (Arts, Science).
**(a)** [1 marks] State the null hypothesis $H_0$ for this test.
**(b)** [1 marks] State the number of degrees of freedom.
**(c)** [4 marks] The calculated test statistic is $\chi^2_{\text{calc}} = 7.48$ with a $p$-value of $0.0237$. At a $5\%$ significance level, state the conclusion of the test with justification.
Verified Solution & Marking Scheme
Part (a) [A1]
$H_0$: Preferred study beverage is independent of faculty (or no association between beverage and faculty).
Part (a) [Note]
Both hypotheses must be stated in context. E.g. $H_0$: Gender and sports preference are independent.
Part (b) [A1]
$\text{df} = (r - 1)(c - 1) = (2 - 1)(3 - 1) = 1 \times 2 = 2$
Part (b) [Note]
Degrees of freedom $\nu = (2-1)(3-1) = 2$. Award A1 for $\chi^2_{\text{calc}}$ to 3 s.f.
Part (c) [R1]
Comparing $p$-value with $\alpha = 0.05$: $0.0237 < 0.05$
Part (c) [A1]
Reject $H_0$ (or evidence suggests significant association).
Part (c) [R1]
There is sufficient evidence at the $5\%$ level to suggest that preferred beverage is dependent on faculty.