In two independent random samples of size [tex]n_1=325[/tex] and [tex]n_2=455[/tex], [tex]\hat{p}_1=0.71[/tex] and [tex]\hat{p}_2=0.64[/tex], calculate the four required quantities for the large-counts condition. If all the counts are at least 10, then the large-counts condition is met.

[tex]
\[
\begin{array}{l}
n_1 \hat{p}_1= \\
n_1\left(1-\hat{p}_1\right)= \\
n_2 \hat{p}_2= \\
n_2\left(1-\hat{p}_2\right)=
\end{array}
\]
[/tex]

To determine if the large-counts condition is met, we need to calculate four quantities and check if each one is at least 10. Let's do it step-by-step:

1. Calculate [tex]\( n_1 \hat{p}_1 \)[/tex]:

- Given [tex]\( n_1 = 325 \)[/tex] and [tex]\( \hat{p}_1 = 0.71 \)[/tex].
- Calculate [tex]\( n_1 \hat{p}_1 = 325 \times 0.71 = 230.75 \)[/tex].

2. Calculate [tex]\( n_1(1 - \hat{p}_1) \)[/tex]:

- Given [tex]\( n_1 = 325 \)[/tex] and [tex]\( \hat{p}_1 = 0.71 \)[/tex].
- Find [tex]\( 1 - \hat{p}_1 = 1 - 0.71 = 0.29 \)[/tex].
- Calculate [tex]\( n_1(1 - \hat{p}_1) = 325 \times 0.29 = 94.25 \)[/tex].

3. Calculate [tex]\( n_2 \hat{p}_2 \)[/tex]:

- Given [tex]\( n_2 = 455 \)[/tex] and [tex]\( \hat{p}_2 = 0.64 \)[/tex].
- Calculate [tex]\( n_2 \hat{p}_2 = 455 \times 0.64 = 291.2 \)[/tex].

4. Calculate [tex]\( n_2(1 - \hat{p}_2) \)[/tex]:

- Given [tex]\( n_2 = 455 \)[/tex] and [tex]\( \hat{p}_2 = 0.64 \)[/tex].
- Find [tex]\( 1 - \hat{p}_2 = 1 - 0.64 = 0.36 \)[/tex].
- Calculate [tex]\( n_2(1 - \hat{p}_2) = 455 \times 0.36 = 163.8 \)[/tex].

Now, let's check if each of these calculated counts is at least 10:

- [tex]\( n_1 \hat{p}_1 = 230.75 \)[/tex] (This is greater than 10)
- [tex]\( n_1(1 - \hat{p}_1) = 94.25 \)[/tex] (This is greater than 10)
- [tex]\( n_2 \hat{p}_2 = 291.2 \)[/tex] (This is greater than 10)
- [tex]\( n_2(1 - \hat{p}_2) = 163.8 \)[/tex] (This is greater than 10)

Since all four quantities are at least 10, the large-counts condition is met.