Question

Primary Heating Fuel. 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.

Suppose that you want to determine whether the distribution of primary heating fuel for occupied housing units built after 2010 differs from that of all occupied housing units. To decide, you take a random sample of housing units built after 2010 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; 300; 400.

c. Strictly speaking, what is the smallest sample size for which conducting a chi-square goodness-of-fit test is appropriate?

Short answer

conducting a chi-square goodness-of-fit test is not appropriate when n = 200 , n= 300 , n= 400 .

Required smallest sample size is 334 for which conducting a chi-square goodness-of-fit test is appropriate.

Step-by-step solution

Step 1. Given 

Primary heating fuel Percentage

Utility gas 50.8

Fuel oil, kerosene 7.9

Electricity 33.9

Bottled, tank, or LPG 5.2

Wood and other fuel 1.9

None 0.3

Step 2. Part ( a )

Identify the number of people

Population:

Population is a complete set of people considered in the study. Here, the population is "all the houses built after 2010"

Find variables in the study

A variable is an attribute or attribute that can be measured. The amount of flexibility may vary for each unit. That is, the variable is defined as the element recorded in each case

From the description given, the variable is the fuel temperature of each unit "because the result of each effort can be measured and this value can vary from effort to individual.

Step 2. Part ( b )

Decide whether to make a chi-square of goodness-of-fit test appropriate or not.

Guess:

• All at least 1 expected frequency.

At least 20% of expected frequencies are less than 5.

• The selected sample should be a simple random sample

Sample size n= 200

The expected frequency is tabulated below :

Primary heating fuel	Percentage	Expected frequency , np
Utility gas	50.8	101.6
fuel oil , kerosene	7.9	15.8
Electricity	33.9	67.8
Bottled tank or LPG	5.2	10.4
Wood and other fuel	1.9	3.8
None	0.3	0.6

From the table, it is observed that the expected frequency for None is less than 1 and 33 %

$\left(=\frac{2}{6}\times 100\right)$of the expected frequencies are less than 5. Therefore, the assumptions are

not satisfied when n=200

Thus, conducting a chi-square goodness-of-fit test is not appropriate when n = 200

Step 4. When sample size , n= 300

Sample size n= 300

The expected frequency is tabulated below :

Primary heating fuel	Percentage	Expected frequency , np
Utility gas	50.8	101.6
fuel oil , kerosene	7.9	15.8
Electricity	33.9	67.8
Bottled tank or LPG	5.2	10.4
Wood and other fuel	1.9	3.8
None	0.3	0.6

From the table, it is observed that the expected frequency for None is less than 1 and 17 %

$\left(=\frac{1}{6}\times 100\right)$of the expected frequencies are less than 5. Therefore, the assumptions are

not satisfied when n=300

Thus, conducting a chi-square goodness-of-fit test is not appropriate when n = 300

Step 5. When sample size , n= 400 .

Sample size n= 400

The expected frequency is tabulated below :

Primary heating fuel	Percentage	Expected frequency , np
Utility gas	50.8	101.6
fuel oil , kerosene	7.9	15.8
Electricity	33.9	67.8
Bottled tank or LPG	5.2	10.4
Wood and other fuel	1.9	3.8
None	0.3	0.6

From the table, it is observed that the expected frequency for None is less than 1 and 17 %

$\left(=\frac{1}{6}\times 100\right)$of the expected frequencies are less than 5. Therefore, the assumptions are

not satisfied when n=400

Thus, conducting a chi-square goodness-of-fit test is not appropriate when n = 400

Step 6. Part ( c ) .

Find the smallest sample size for which conducting a chi-square goodness-of-fit test is appropriate.

The required expected frequency for "None" should be greater than or equal to 1. That is , $\mathrm{n}\left(\frac{0.3}{100}\right)>1$Therefore, the smallest sample size for which conducting a chi-square goodness of-fit test is appropriate is,

n(0.003) = 1

$\mathrm{n}=\frac{1}{0.003}=333.33\approx 334$

Thus, the required smallest sample size is 334