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

Answer :

To determine if the large-counts condition is met, we need to calculate four specific quantities using the given sample sizes and sample proportions. Here’s how each quantity is calculated step-by-step:

1. Calculate [tex]\( n_1 \hat{\beta}_1 \)[/tex]:
- We have the first sample size [tex]\( n_1 = 325 \)[/tex].
- The first sample proportion [tex]\( \hat{\beta}_1 = 0.71 \)[/tex].
- Multiply these two values to find the expected count of successes:
[tex]\[ n_1 \hat{\beta}_1 = 325 \times 0.71 = 230.75 \][/tex]

2. Calculate [tex]\( n_1(1 - \hat{\beta}_1) \)[/tex]:
- The first sample’s proportion of failures is [tex]\( 1 - \hat{\beta}_1 = 1 - 0.71 = 0.29 \)[/tex].
- Multiply by the sample size to find the expected count of failures:
[tex]\[ n_1(1 - \hat{\beta}_1) = 325 \times 0.29 = 94.25 \][/tex]

3. Calculate [tex]\( n_2 \hat{\beta}_2 \)[/tex]:
- For the second sample, the size is [tex]\( n_2 = 455 \)[/tex].
- The second sample proportion [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Multiply these to find the expected count of successes:
[tex]\[ n_2 \hat{\beta}_2 = 455 \times 0.64 = 291.2 \][/tex]

4. Calculate [tex]\( n_2(1 - \hat{\beta}_2) \)[/tex]:
- The second sample’s proportion of failures is [tex]\( 1 - \hat{\beta}_2 = 1 - 0.64 = 0.36 \)[/tex].
- Multiply by the sample size to find the expected count of failures:
[tex]\[ n_2(1 - \hat{\beta}_2) = 455 \times 0.36 = 163.8 \][/tex]

Now, to determine if the large-counts condition is met, we check whether all calculated counts are at least 10. Here are the results:

- [tex]\( n_1 \hat{\beta}_1 = 230.75 \)[/tex]
- [tex]\( n_1(1 - \hat{\beta}_1) = 94.25 \)[/tex]
- [tex]\( n_2 \hat{\beta}_2 = 291.2 \)[/tex]
- [tex]\( n_2(1 - \hat{\beta}_2) = 163.8 \)[/tex]

All these values are greater than 10, so the large-counts condition is met.

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