To establish the null and alternative hypotheses for the conjecture "The majority of college students have credit cards," we need to define what we mean by "the majority." Typically, "majority" implies more than 50%.
Here’s how you can formulate the hypotheses:
Null Hypothesis (H0)
The null hypothesis represents the statement we assume to be true until we have sufficient evidence to suggest otherwise. In this case, it can be stated as:
- \( H_0: p \leq 0.5 \) (The proportion of college students with credit cards is 50% or less.)
Alternative Hypothesis (H1)
The alternative hypothesis represents the statement that we are trying to find evidence for. In this context, it can be stated as:
- \( H_1: p > 0.5 \) (The proportion of college students with credit cards is more than 50%.)
In summary:
a) The null hypothesis is \( H_0: p \leq 0.5 \)
b) The alternative hypothesis is \( H_1: p > 0.5 \)