Question

Population by Region. According to the U.S. Census Bureau publication Demographic Profiles, a relative-frequency distribution of the U.S. resident population by region in 2000 was as follows.

A simple random sample of this year's U.S. residents gave the following frequency distribution.

a. Identify the population and variable under consideration here.

b. At the $5\%$ significance level, do the data provide sufficient evidence to conclude that this year's resident population distribution by region has changed from the $2000$ distribution?

Short answer

a). The variable is Region, and the population is the population of the United States.

b). At a $5\%$ significance level, there is enough information to establish that the resident population distribution by area this year differs from the resident population distribution in $2000$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

A simple random sample of this year's U.S. residents gave the following frequency distribution.

Part (a) Step 2: Explanation

Individuals who are being examined form the population. As a result, only U.S. residents are taken into account.

The subject of investigation is the variable. As a result, the region is the factor to consider.

Part (b) Step 1: Given Information

Given data:

A simple random sample of this year's U.S. residents gave the following frequency distribution.

Part (b) Step 2: Explanation

The hypotheses:

$H_0$: The resident population distribution by region for this year is same to the resident population distribution for the year $2000$.

$H_a$: The resident population distribution by region for this year differs from the resident population distribution for the year $2000$.

$\alpha =5\%$

$=0.05$

Calculate the observed frequencies and chi-square subtotals as follows:

localid="1653750249413" $\begin{array}{|llll|}\hline p& O& E=1500p& (O-E{)}^2/E\\ 0.19& 259& 285& 2.3719\\ 0.229& 306& 343.5& 4.0939\\ 0.356& 568& 534& 2.1648\\ 0.225& 367& 337.5& 2.5785\\ \text{Sum}& 1500& & 11.2091\\ \hline\end{array}$

${\chi }^2=11.2091$

Determine the critical value using Table $\mathrm{V}$with $df=4-1$

$=3$:

${\chi }_{0.05}^2=7.815$

$11.2091>7.815\Rightarrow \text{Reject}{H}_0$