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Answer :
To determine whether the large counts condition is met for constructing a confidence interval for the proportion [tex]\( p \)[/tex], we need to check if both [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1 - \hat{p}) \)[/tex] are at least 10. Let's break it down step by step:
1. Calculate [tex]\( n \hat{p} \)[/tex]:
The formula for this calculation is [tex]\( n \times \hat{p} \)[/tex].
Given that [tex]\( n = 50 \)[/tex] and [tex]\( \hat{p} = 0.9 \)[/tex], we perform the following calculation:
[tex]\( 50 \times 0.9 = 45 \)[/tex].
2. Calculate [tex]\( n(1 - \hat{p}) \)[/tex]:
This formula is [tex]\( n \times (1 - \hat{p}) \)[/tex].
Again, using [tex]\( n = 50 \)[/tex] and [tex]\( \hat{p} = 0.9 \)[/tex], the calculation is:
[tex]\( 50 \times (1 - 0.9) = 50 \times 0.1 = 5 \)[/tex].
3. Check each condition:
- For [tex]\( n \hat{p} \)[/tex]: We have [tex]\( 45 \)[/tex], which is greater than 10. This condition is satisfied.
- For [tex]\( n(1 - \hat{p}) \)[/tex]: We have [tex]\( 5 \)[/tex], which is not greater than or equal to 10. This condition is not satisfied.
Since not both conditions are satisfied, the large counts condition is not met. Therefore, the correct answer is:
No, [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1 - \hat{p}) \)[/tex] are not both at least 10.
1. Calculate [tex]\( n \hat{p} \)[/tex]:
The formula for this calculation is [tex]\( n \times \hat{p} \)[/tex].
Given that [tex]\( n = 50 \)[/tex] and [tex]\( \hat{p} = 0.9 \)[/tex], we perform the following calculation:
[tex]\( 50 \times 0.9 = 45 \)[/tex].
2. Calculate [tex]\( n(1 - \hat{p}) \)[/tex]:
This formula is [tex]\( n \times (1 - \hat{p}) \)[/tex].
Again, using [tex]\( n = 50 \)[/tex] and [tex]\( \hat{p} = 0.9 \)[/tex], the calculation is:
[tex]\( 50 \times (1 - 0.9) = 50 \times 0.1 = 5 \)[/tex].
3. Check each condition:
- For [tex]\( n \hat{p} \)[/tex]: We have [tex]\( 45 \)[/tex], which is greater than 10. This condition is satisfied.
- For [tex]\( n(1 - \hat{p}) \)[/tex]: We have [tex]\( 5 \)[/tex], which is not greater than or equal to 10. This condition is not satisfied.
Since not both conditions are satisfied, the large counts condition is not met. Therefore, the correct answer is:
No, [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1 - \hat{p}) \)[/tex] are not both at least 10.
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