Question

a survey of 36 randomly selected students who dropped a course was conducted at a college. the following results were collected. complete parts (a) through (c).
full data set
f personal | m course | m course
m work | f personal | m course
f work | m work | m personal
m personal | f course | f course
m course | f personal | m course
m personal | f personal | m course
m course | f work | m course
f personal | f course | f work
f personal | f course | m work
f personal | f work | m course
f work | m work | f personal
f personal | f personal | m work

(a) construct a contingency table for the two variables.
course | personal | work
male | 9 | 3 | 5
female | 4 | 10 | 5

(b) test whether sex is independent of drop reason at the \\(\alpha = 0.1\\) level of significance.
what are the hypotheses?
a. \\(h_0: o_c = e_c\\) and \\(o_p = e_p\\) and \\(o_w = e_w\\)
\\(h_1: o_c \
eq e_c\\) or \\(o_p \
eq e_p\\) or \\(o_w \
eq e_w\\)
b. \\(h_0: p_c = p_p = p_w\\)
\\(h_1\\): at least one of the proportions is different from the others.
c. \\(h_0\\): sex and drop reason are independent.
\\(h_1\\): sex and drop reason are dependent.

Explanation:

Identify the given data and variables

The problem involves two categorical variables: sex (Male, Female) and drop reason (Course, Personal, Work). Part (a) shows a completed Contingency Table summarizing the frequencies. Part (b) asks to test whether sex is independent of drop reason.

State the hypotheses for independence

For a Chi-Square Test of Independence, the null hypothesis \(H_0\) always states that the two categorical variables are independent. The alternative hypothesis \(H_1\) states that the two variables are dependent (not independent).

Match with the given options

• Option A states hypotheses in terms of observed and expected frequencies, which is incorrect for defining independence.
• Option B states hypotheses in terms of equal proportions, which is used for testing homogeneity or equal proportions across populations, not independence of two variables.
• Option C states:

\(H_0\): Sex and drop reason are independent.
\(H_1\): Sex and drop reason are dependent.
This matches the standard definition for a test of independence.

Answer:

Question 1

• (A) \(H_0: O_C = E_C \text{ and } O_P = E_P \text{ and } O_W = E_W\)
• (B) \(H_0: p_C = p_P = p_W\)
• (C) \(H_0\): Sex and drop reason are independent. \(H_1\): Sex and drop reason are dependent. (Correct answer)