Problem Statement
A university surveyed 200 students regarding their preferred study location (Library, Cafe, Home) across their major (STEM, Humanities):
$$\begin{array}{r|ccc|c} & \text{Library} & \text{Cafe} & \text{Home} & \text{Total} \\ \hline \text{STEM} & 55 & 25 & 30 & 110 \\ \text{Humanities} & 25 & 35 & 30 & 90 \\ \hline \text{Total} & 80 & 60 & 60 & 200 \end{array}$$
A $\chi^2$ test of independence is conducted at the $5\%$ significance level.
**(a)** [1 marks] State the null hypothesis $H_0$ for this test.
**(b)** [1 marks] Calculate the expected frequency of STEM students who prefer studying in a Cafe.
**(c)** [2 marks] State the degrees of freedom and find the $p$-value for this test using your GDC.
**(d)** [2 marks] State the conclusion of the test in context, justifying your answer.
Verified Solution & Marking Scheme
Part (a) [A1]
$H_0$: Preferred study location is independent of academic major.
Part (b) [A1]
$E = \frac{\text{Row Total} \times \text{Col Total}}{\text{Grand Total}} = \frac{110 \times 60}{200} = 33$
Part (c) [A1]
$\text{df} = (r - 1)(c - 1) = (2 - 1)(3 - 1) = 1 \times 2 = 2$
Part (c) [A1]
From GDC 2-way $\chi^2$ test: $\chi^2_{\text{calc}} \approx 9.76, \quad p\text{-value} \approx 0.00759$
Part (d) [R1]
Since $p\text{-value} = 0.00759 < 0.05$ (or $\chi^2_{\text{calc}} = 9.76 > 5.991$), we reject $H_0$.
Part (d) [A1]
There is sufficient evidence at the $5\%$ significance level to conclude that study location preference is dependent on academic major.