June is a researcher who read a 2016 study that published the following population distribution for Americans:

[tex]
\[
\begin{tabular}{lrrrrrr}
Age group & $0-18$ & $19-25$ & $26-34$ & $35-54$ & $55-64$ & $65+$ \\
\hline
Percentage & $24\%$ & $9\%$ & $12\%$ & $26\%$ & $13\%$ & $15\%$ \\
\end{tabular}
\]
[/tex]

She wonders if these figures still hold true, so she takes a sample of 38 Americans and records their ages. Here are the results:

[tex]
\[
\begin{tabular}{lrrrrrr}
Age group & $0-18$ & $19-25$ & $26-34$ & $35-54$ & $55-64$ & $65+$ \\
\hline
Observed counts & 9 & 3 & 5 & 10 & 5 & 6 \\
\end{tabular}
\]
[/tex]

June wants to use these results to carry out a [tex]\chi^2[/tex] goodness-of-fit test to determine if her sample disagrees with the official percentages.

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

Choose 3 answers:

A. The observed count of the 19-25 age group.

B. The observed count of the 26-34 age group.

C. The expected count of the 19-25 age group.

D. The expected count of the 26-34 age group.

E. The expected count of the 55-64 age group.

To perform a [tex]$\chi^2$[/tex] goodness-of-fit test, one of the conditions is that the expected counts in each category should be at least 5. Given a sample of 38 Americans and the official percentages, the expected count for an age group is calculated as

[tex]$$
\text{Expected Count} = (\text{Sample Size}) \times (\text{Percentage})
$$[/tex]

For each age group:

1. For the [tex]$0-18$[/tex] group:
[tex]$$
38 \times 0.24 = 9.12
$$[/tex]
This is greater than 5.

2. For the [tex]$19-25$[/tex] group:
[tex]$$
38 \times 0.09 = 3.42
$$[/tex]
This expected count is less than 5.

3. For the [tex]$26-34$[/tex] group:
[tex]$$
38 \times 0.12 = 4.56
$$[/tex]
This expected count is less than 5.

4. For the [tex]$35-54$[/tex] group:
[tex]$$
38 \times 0.26 = 9.88
$$[/tex]
This is greater than 5.

5. For the [tex]$55-64$[/tex] group:
[tex]$$
38 \times 0.13 = 4.94
$$[/tex]
This expected count is less than 5.

6. For the [tex]$65+$[/tex] group:
[tex]$$
38 \times 0.15 = 5.70
$$[/tex]
This is greater than 5.

Since the expected counts for the [tex]$19-25$[/tex], [tex]$26-34$[/tex], and [tex]$55-64$[/tex] age groups are less than 5, these categories fail the large counts condition.

Thus, the counts that cause the failure of the large counts condition are in the [tex]$19-25$[/tex], [tex]$26-34$[/tex], and [tex]$55-64$[/tex] groups.

The correct answer choices are:

C, D, and E.