Question

the conditional relative frequency table was generated by row using frequency table data comparing the hat size and shirt size of children on a baseball team.
hat size and shirt size

	child - sized hat	adult - sized hat	total
large shirt	0.2	0.8	1.0
total	0.48	0.52	1.0

the coach attempts to determine an association between shirt size and hat size. which is most likely true?

• an association cannot be determined because 0.48 is similar to 0.52.
• an association cannot be determined because the sum of each column is not 1.0.
• there is likely an association because 0.8 is not similar to 0.2.
• there is likely an association because 0.8 is not similar to ≈0.33.

Explanation:

To determine an association between shirt size and hat size using a conditional relative frequency table (generated by row), we compare the conditional relative frequencies within each row (shirt size group) for the different hat size categories.

Step 1: Analyze the rows (shirt sizes)

• Medium Shirt row: Child - sized Hat ≈ 0.67, Adult - sized Hat ≈ 0.33.
• Large Shirt row: Child - sized Hat = 0.2, Adult - sized Hat = 0.8.

Step 2: Evaluate association

For an association to exist, the conditional relative frequencies of hat sizes should differ significantly between the shirt size groups.

• In the Medium Shirt group, the proportion of children with Adult - sized Hats is ≈ 0.33. In the Large Shirt group, the proportion of children with Adult - sized Hats is 0.8. These two proportions (0.8 and ≈0.33) are not similar, indicating a difference in hat size distribution between the two shirt size groups.
• The first option is incorrect because 0.48 and 0.52 are marginal totals (column totals) and not relevant for determining association within rows.
• The second option is incorrect because the table is generated by row, so we expect row totals to be 1.0 (which they are), and column totals do not need to be 1.0 for a row - generated conditional relative frequency table.
• The third option compares 0.8 and 0.2, which are the proportions of Adult - sized and Child - sized Hats within the Large Shirt group, not across groups. The key is to compare the same hat size proportion across different shirt size groups.

Answer:

There is likely an association because 0.8 is not similar to ≈ 0.33.