Question

in a previous year, 54% of women aged 15 and older lived alone. a sociologist tests whether this percentage is different today by conducting a random sample of 600 women aged 15 and older and finds that 330 are living alone. is there sufficient evidence at the \\( \alpha = 0.05 \\) level of significance to conclude the proportion has changed?
determine whether all of requirements for the hypothesis test are satisfied. complete the following sentence
because \\( n p _ { 0 } ( 1 - p _ { 0 } ) = \square \\) 10, the sample size is 5% of the population size, and the sample a random sample, all of the requirements for testing the hypothesis satisfied
(round to one decimal place as needed.)

Explanation:

Step1: Calculate \(np_0(1 - p_0)\)

Given \(n = 600\), \(p_0=0.54\). Then \(np_0(1 - p_0)=600\times0.54\times(1 - 0.54)\)

$$
LATEXBLOCK0
$$

Since \(149.0> 10\)

Step2: Check sample - size relative to population size

We assume that the sample of \(n = 600\) women is a random sample. Also, unless we are told otherwise about the population size of women aged 15 and older (which is very large, say in millions), \(n = 600\) is less than \(5\%\) of the population size.

Answer:

Because \(np_0(1 - p_0)=149.0>10\), the sample size is less than \(5\%\) of the population size, and the sample is a random sample, all of the requirements for testing the hypothesis are satisfied.