Terrell's company sells candy in packs that are supposed to contain 50% red candies, 25% orange, and 25% yellow. He randomly selected a pack containing 16 candies and counted how many of each color were in the pack. Here are his results:

[tex]
\[
\begin{tabular}{|l|r|r|r|}
\hline
\text{Color} & \text{Red} & \text{Orange} & \text{Yellow} \\
\hline
\text{Observed counts} & 9 & 5 & 2 \\
\hline
\end{tabular}
\]
[/tex]

He wants to use these results to carry out a [tex]\(\chi^2\)[/tex] goodness-of-fit test to determine if the color distribution disagrees with the target percentages.

Which count(s) make this sample fail the large counts condition for this test?

To determine if the sample fails the large counts condition for the chi-square goodness-of-fit test, we'll follow these steps:

1. Understand the Expected Percentages: The candies are supposed to be distributed as 50% red, 25% orange, and 25% yellow.

2. Calculate the Expected Counts: Since the total number of candies is 16:

- Expected count for red candies: [tex]\( 16 \times 0.5 = 8 \)[/tex]
- Expected count for orange candies: [tex]\( 16 \times 0.25 = 4 \)[/tex]
- Expected count for yellow candies: [tex]\( 16 \times 0.25 = 4 \)[/tex]

3. Compare with the Observed Counts: The observed counts for each color were:

- Red: 9
- Orange: 5
- Yellow: 2

4. Check the Large Counts Condition: This condition requires that all expected counts should be at least 5 to ensure the validity of the chi-square test.

- The expected count for red candies is 8 (which is 5 or more).
- The expected count for orange candies is 4 (which is less than 5).
- The expected count for yellow candies is 4 (which is less than 5).

5. Conclusion: The expected counts for orange and yellow candies (both being 4) make this sample fail the large counts condition, because they are less than 5. Hence, the sample does not adequately meet the conditions for a chi-square goodness-of-fit test due to these smaller expected counts.