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Answer :
To determine if the large counts condition is met for constructing a confidence interval for [tex]\( p \)[/tex], we need to consider the values of [tex]\( n \)[/tex], the sample size, and [tex]\(\hat{p}\)[/tex], the sample proportion. In this problem, the values given are [tex]\( n = 50 \)[/tex] and [tex]\(\hat{p} = 0.9\)[/tex].
Step 1: Calculate [tex]\( n \hat{p} \)[/tex].
We multiply the sample size [tex]\( n \)[/tex] by the sample proportion [tex]\(\hat{p}\)[/tex]:
[tex]\[ n \hat{p} = 50 \times 0.9 = 45 \][/tex]
Step 2: Calculate [tex]\( n (1 - \hat{p}) \)[/tex].
Next, we calculate the product of the sample size [tex]\( n \)[/tex] and the complement of the sample proportion [tex]\( (1 - \hat{p}) \)[/tex]:
[tex]\[ n (1 - \hat{p}) = 50 \times (1 - 0.9) = 50 \times 0.1 = 5 \][/tex]
Step 3: Check the large counts condition.
The large counts condition requires both [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] to be at least 10 for the condition to be met.
- For [tex]\( n \hat{p} = 45 \)[/tex], which is greater than 10, the condition is satisfied.
- For [tex]\( n(1 - \hat{p}) = 5 \)[/tex], which is less than 10, the condition is not satisfied.
Since both conditions must be met and [tex]\( n(1 - \hat{p}) \)[/tex] is not at least 10, the large counts condition is not met.
Therefore, the correct conclusion is:
No, [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] are not both at least 10.
Step 1: Calculate [tex]\( n \hat{p} \)[/tex].
We multiply the sample size [tex]\( n \)[/tex] by the sample proportion [tex]\(\hat{p}\)[/tex]:
[tex]\[ n \hat{p} = 50 \times 0.9 = 45 \][/tex]
Step 2: Calculate [tex]\( n (1 - \hat{p}) \)[/tex].
Next, we calculate the product of the sample size [tex]\( n \)[/tex] and the complement of the sample proportion [tex]\( (1 - \hat{p}) \)[/tex]:
[tex]\[ n (1 - \hat{p}) = 50 \times (1 - 0.9) = 50 \times 0.1 = 5 \][/tex]
Step 3: Check the large counts condition.
The large counts condition requires both [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] to be at least 10 for the condition to be met.
- For [tex]\( n \hat{p} = 45 \)[/tex], which is greater than 10, the condition is satisfied.
- For [tex]\( n(1 - \hat{p}) = 5 \)[/tex], which is less than 10, the condition is not satisfied.
Since both conditions must be met and [tex]\( n(1 - \hat{p}) \)[/tex] is not at least 10, the large counts condition is not met.
Therefore, the correct conclusion is:
No, [tex]\( n \hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] are not both at least 10.
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