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Answer :
To determine whether the large counts condition is met for constructing a confidence interval for [tex]\( p \)[/tex], we need to check two things:
1. [tex]\( n \hat{p} \)[/tex] must be at least 10.
2. [tex]\( n(1 - \hat{p}) \)[/tex] must be at least 10.
Given:
- [tex]\( n = 50 \)[/tex]
- [tex]\( \hat{p} = 0.9 \)[/tex]
Let's calculate each part:
1. Calculate [tex]\( n \hat{p} \)[/tex]:
[tex]\[
n \hat{p} = 50 \times 0.9 = 45
\][/tex]
2. Calculate [tex]\( n(1 - \hat{p}) \)[/tex]:
[tex]\[
n(1 - \hat{p}) = 50 \times (1 - 0.9) = 50 \times 0.1 = 5
\][/tex]
Now, check if these values meet the condition:
- [tex]\( n \hat{p} = 45 \)[/tex] is greater than or equal to 10. This condition is satisfied.
- [tex]\( n(1 - \hat{p}) = 5 \)[/tex] is less than 10. This condition is not satisfied.
Since both conditions must be met for the large counts condition to hold, and only one is satisfied here, the large counts condition is not met.
Therefore, the answer is:
No, [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1 - \hat{p}) \)[/tex] are not both at least 10.
1. [tex]\( n \hat{p} \)[/tex] must be at least 10.
2. [tex]\( n(1 - \hat{p}) \)[/tex] must be at least 10.
Given:
- [tex]\( n = 50 \)[/tex]
- [tex]\( \hat{p} = 0.9 \)[/tex]
Let's calculate each part:
1. Calculate [tex]\( n \hat{p} \)[/tex]:
[tex]\[
n \hat{p} = 50 \times 0.9 = 45
\][/tex]
2. Calculate [tex]\( n(1 - \hat{p}) \)[/tex]:
[tex]\[
n(1 - \hat{p}) = 50 \times (1 - 0.9) = 50 \times 0.1 = 5
\][/tex]
Now, check if these values meet the condition:
- [tex]\( n \hat{p} = 45 \)[/tex] is greater than or equal to 10. This condition is satisfied.
- [tex]\( n(1 - \hat{p}) = 5 \)[/tex] is less than 10. This condition is not satisfied.
Since both conditions must be met for the large counts condition to hold, and only one is satisfied here, the large counts condition is not met.
Therefore, the 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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