Terrell's company sells candy in packs that are supposed to contain [tex]$50 \%$[/tex] red candies, [tex]$25 \%$[/tex] orange, and [tex]$25 \%$[/tex] 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{array}{lrrr}

\text{Color} & \text{Red} & \text{Orange} & \text{Yellow} \\

\hline \text{Observed counts} & 9 & 5 & 2

\end{array}

\][/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?

Choose 2 answers:

A. The observed count of yellow candies.
B. The observed count of orange candies.
C. The expected count of red candies.
D. The expected count of orange candies.
E. The expected count of yellow candies.

To determine if Terrell's sample fails the large counts condition for the chi-squared goodness-of-fit test, follow these steps:

1. Determine the Expected Counts:
- The pack contains a total of 16 candies. According to the target percentages:
- Red candies should be 50% of the pack, so expected count is [tex]$ 16 \times 0.50 = 8 $[/tex].
- Orange candies should be 25% of the pack, so expected count is [tex]$ 16 \times 0.25 = 4 $[/tex].
- Yellow candies should be 25% of the pack, so expected count is [tex]$ 16 \times 0.25 = 4 $[/tex].

2. Check the Large Counts Condition:
- The large counts condition requires that all expected counts should be at least 5 for the chi-squared test to be valid.
- The expected count for red candies is 8, which is greater than 5, so it satisfies the large counts condition.
- The expected count for orange candies is 4, which is less than 5, so it does not satisfy the large counts condition.
- The expected count for yellow candies is 4, which is also less than 5, so it does not satisfy the large counts condition.

3. Determine Which Counts Fail the Condition:
- The expected counts for both orange and yellow candies fail the large counts condition as they are less than 5.

Thus, the correct answers are:
- D: The expected count of orange candies.
- E: The expected count of yellow candies.