Chapter 11: Q. 49 (page 756)

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

Expert verified

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