According to Current Housing Reports, published by the U.S....

According to Current Housing Reports, published by the U.S. Census Bureau, the primary heating fuel for all occupied housing units is distributed as follows. $$\begin{array}{|l|c|} \hline \text { Primary heating fuel } & \text { Percentage } \\ \hline \text { Utility gas } & 51.5 \\ \text { Fuel oil, kerosene } & 9.8 \\ \text { Electricity } & 30.7 \\ \text { Bottled, tank, or LPG } & 5.7 \\ \text { Wood and other fucl } & 1.9 \\ \text { None } & 0.4 \\ \hline \end{array}$$ Suppose that you want to determine whether the distribution of primary heating fuel for occupied housing units built after 2000 differs from that of all occupied housing units. To decide, you take a random sample of housing units built after 2000 and obtain a frequency distribution of their primary heating fuel. a. Identify the population and variable under consideration here. b. For each of the following sample sizes, determine whether conducting a chi-square goodness-of-fit test is appropriate and explain your answers: $200 ; 250 ; 300$. c. Strictly speaking, what is the smallest sample size for which conducting a chi-square goodness-of-fit test is appropriate?

According to Current Housing Reports, published by the U.S. Census Bureau, the primary heating fuel for all occupied housing units is distributed as follows.
$$\begin{array}{|l|c|}
\hline \text { Primary heating fuel } & \text { Percentage } \\
\hline \text { Utility gas } & 51.5 \\
\text { Fuel oil, kerosene } & 9.8 \\
\text { Electricity } & 30.7 \\
\text { Bottled, tank, or LPG } & 5.7 \\
\text { Wood and other fucl } & 1.9 \\
\text { None } & 0.4 \\
\hline
\end{array}$$
Suppose that you want to determine whether the distribution of primary heating fuel for occupied housing units built after 2000 differs from that of all occupied housing units. To decide, you take a random sample of housing units built after 2000 and obtain a frequency distribution of their primary heating fuel.
a. Identify the population and variable under consideration here.
b. For each of the following sample sizes, determine whether conducting a chi-square goodness-of-fit test is appropriate and explain your answers: $200 ; 250 ; 300$.
c. Strictly speaking, what is the smallest sample size for which conducting a chi-square goodness-of-fit test is appropriate?

Key Concepts

Expected Frequency Condition

The requirement for the expected frequency in each category—usually at least 5—is critical when deciding whether a chi-square test is appropriate. If the sample size is too small, some categories might have expected frequencies below this threshold, undermining the approximation to the chi-square distribution. This concept guides analysts in determining the minimal sample size needed to reliably conduct the test.

Chi-square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical procedure used to determine if an observed frequency distribution deviates significantly from an expected distribution. This test compares the observed counts in each category to the counts that would be expected under a specified theoretical distribution, making it a useful tool in assessing categorical data.

Assumptions of the Chi-square Test

This test has important assumptions that must be met: observations should be independent and randomly sampled, categories must be mutually exclusive, and the expected frequency in each category should be sufficiently large (typically at least 5). These assumptions ensure the test’s validity and reliability when making statistical inferences.

Population

The population refers to the full set of subjects or items about which conclusions are to be drawn. In the context of testing the distribution of primary heating fuels, the population is the subgroup of housing units built after 2000. More generally, understanding the population is crucial in statistical inference because any conclusion drawn from a sample is generalized back to this larger group.

Variable

A variable is a characteristic or measurement that can take on different values among the items in the population. Here, the variable of interest is the type of primary heating fuel used by a housing unit. In statistical studies, correctly identifying the variable ensures that the data collected are appropriate for addressing the research question.

Transcript

Please give Ace some feedback