91Ó°ÊÓ

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

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.

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 :

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

$\left(=\frac{2}{6}×100\right)$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

Sample size n= 300

The expected frequency is tabulated below :

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

$\left(=\frac{1}{6}×100\right)$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

Sample size n= 400

The expected frequency is tabulated below :

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

$\left(=\frac{1}{6}×100\right)$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

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

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

Thus, the required smallest sample size is 334