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------------------------------------------------ Suppose we want to construct a confidence interval for [tex] p [/tex] with [tex] n = 50 [/tex] and [tex] \hat{p} = 0.9 [/tex]. Is the large counts condition met?

A. Yes, [tex] n \hat{p} [/tex] is at least 10.
B. Yes, [tex] n(1-\hat{p}) [/tex] is at least 10.
C. Yes, both [tex] n \hat{p} [/tex] and [tex] n(1-\hat{p}) [/tex] are at least 10.
D. No, [tex] n \hat{p} [/tex] and [tex] n(1-\hat{p}) [/tex] are not both at least 10.

To determine if the large counts condition is met for constructing a confidence interval for a proportion [tex]\( p \)[/tex], we need to check two conditions:

1. [tex]\( n \hat{p} \)[/tex] should be at least 10.
2. [tex]\( n(1 - \hat{p}) \)[/tex] should be at least 10.

Given Values:

- [tex]\( n = 50 \)[/tex] (sample size)
- [tex]\( \hat{p} = 0.9 \)[/tex] (sample proportion)

Step 1: Calculate [tex]\( n \hat{p} \)[/tex].

[tex]\[ n \hat{p} = 50 \times 0.9 = 45 \][/tex]

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

[tex]\[ 1 - \hat{p} = 1 - 0.9 = 0.1 \][/tex]

[tex]\[ n(1 - \hat{p}) = 50 \times 0.1 = 5 \][/tex]

Analysis:

1. [tex]\( n \hat{p} = 45 \)[/tex] which is greater than 10, so this condition is satisfied.
2. [tex]\( n(1 - \hat{p}) = 5 \)[/tex] which is less than 10, so this condition is not satisfied.

Conclusion:

Since both conditions need to be met to say the large counts condition is satisfied and only the first condition is met, the large counts condition is not met in this case. Therefore, the answer is:

No, [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1 - \hat{p}) \)[/tex] are not both at least 10.