Null Hypothesis formulation from a "Fewer than" statement

Question

The aim of this exercise is to translate statements into Null and Alternative Hypotheses.

The example statement is: "Fewer than 5% of adults ride the bus to work in Los Angeles."

My Answer was : 0: ≤0.05;: >0.05.

• Given that 0 always has a symbol of =, ≤, or ≥.
• Given that always has a symbol of >, <, ≠.

However, the correct answer is : 0: =0.05;: <0.05

1. Why is the 0 statement 0: =0.05 and not : >0.05 ?
2. Why is the statement : <0.05 and not : >0.05 ?

Answer 1

"Fewer than 5% of adults ride the bus to work in Los Angeles."

The null hypothesis to be verified is the complement: $H_0\geq 5\%$

To set up a test, we expect that the data contraddict this Hypothesis, i.e. sample mean is $4\%$. The test is performed to tell us if actually $4<5$ or the difference has to be attributed to the phenomena's variability.

The system is the following

$$\begin{cases} H_0: p\geq 5\%\\ H_1: p<5\% \end{cases}$$

This is a good answer and correct too.

For some reasons that should be clear to you, the previous Hypothesis System is equivalent to the following one

$$\begin{cases} H_0: p= 5\%\\ H_1: p<5\% \end{cases}$$