Question

Where do young adults live? A survey by the National Institutes of Health asked a random sample of young adults (aged $19$to $25$years), “Where do you live now? That is, where do you stay most often?” Here is the full two-way table (omitting a few who refused to answer and one who reported being homeless):

a. Should we use a chi-square test for homogeneity or a chi-square test for independence in this setting? Justify your answer.

b. State appropriate hypotheses for performing the type of test you chose in part (a). Here is Minitab output from a chi-square test.

c. Check that the conditions for carrying out the test are met.

d. Interpret the P-value. What conclusion would you draw?

Short answer

a. We should use Chi-square test for independence.

b. $H_0$: Gender has nothing to do with where you reside.

$H_a$:Gender and place of residence are linked.

c. The chi square test has three conditions: randomness, independence, and large counts. Because the sample was chosen at random, the random sample condition was met.

d. There is convincing evidence that the association between gender and living location

Step-by-step solution

Part (a) Step 1 : Given Information

We have to determine which chi-square test is appropriate for given setting.

Part (a) Step 2 : Simplification

First, we must determine which test should be used in a certain case. There are three tests to complete:

Chi-square goodness-of-fit test, chi-square homogeneity test, and chi-square independence test All of these tests will be detailed for us. A chi-square goodness-of-fit test is used when we are interested in the distribution of a single variable. In this circumstance, we must employ a chi-square test for homogeneity when we are interested in the distribution of two variables with numerous independent samples. We would like to do a chi-square test for independence when we are interested in the distribution of two variables and there is only one sample. We are given two variables for a certain setting: Gender and place of residence There is only one random sample available.

Asaresult,weshouldperformtheChi-squaretesttodetermineindependence.

Part (b) Step 1 : Given Information

We have to state the null and alternative hypotheses.

Part (b) Step 2 : Simplification

The following are the null and alternate hypotheses:
$H_0$: Gender has nothing to do with where you reside.

$H_a$: Gender and where you reside have a connection.

Part (c) Step 1 : Given Information

We have to verify the conditions for inference.

Part (c) Step 2 : Simplification

Thechisquaretesthasthreeconditions:randomness,independence,andlargecounts.

Because the sample was chosen at random, the random sample condition was met.

Because the sample size is less than $10\%$of the total population, the requirement of independence is met.

Because each survey's projected count is at least $5$, the criterion of a large count is met.

Part (d) Step 1 : Given Information

We have to explain conclusion.

Part (d) Step 2 : Simplification

The p-value = $0.012$. If there is no relationship between the two variables, the probability of getting a sample that is similar to or more extreme than the given sample is $0.012$or $1.2$percent.$H_0$is rejected because the P-value is less than $0.05$.

Conclusion: There is strong evidence that there is a link between gender and where you live.