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------------------------------------------------ Understanding the Conditions

Suppose we want to construct a confidence interval for [tex] p [/tex] and [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, we need to look at two calculations using the sample size [tex]\( n \)[/tex] and the probability [tex]\( B \)[/tex].

1. Calculate [tex]\( nB \)[/tex]:
- Multiply the sample size [tex]\( n = 50 \)[/tex] by the probability [tex]\( B = 0.9 \)[/tex].
- Calculation: [tex]\( nB = 50 \times 0.9 = 45 \)[/tex].

2. Calculate [tex]\( n(1-B) \)[/tex]:
- First find [tex]\( 1-B \)[/tex], which is [tex]\( 1 - 0.9 = 0.1 \)[/tex].
- Then multiply by the sample size: [tex]\( n(1-B) = 50 \times 0.1 = 5 \)[/tex].

3. Determine if the large counts condition is met:
- For the large counts condition, both [tex]\( nB \)[/tex] and [tex]\( n(1-B) \)[/tex] need to be at least 10.
- Here, [tex]\( nB = 45 \)[/tex], which is at least 10.
- However, [tex]\( n(1-B) = 5 \)[/tex], which is less than 10.

Since both values are not at least 10, the large counts condition is not met. Therefore, the answer is:
No, [tex]\( nB \)[/tex] and [tex]\( n(1-B) \)[/tex] are not both at least 10.