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Answer :
To determine whether the large counts condition is met for constructing a confidence interval for the population proportion [tex]\( p \)[/tex], we need to check the following two criteria:
1. [tex]\( n \hat{p} \geq 10 \)[/tex]
2. [tex]\( n(1 - \hat{p}) \geq 10 \)[/tex]
Here's how you can solve this:
1. Compute [tex]\( n \hat{p} \)[/tex]:
- Given:
- Sample size [tex]\( n = 50 \)[/tex]
- Sample proportion [tex]\( \hat{p} = 0.9 \)[/tex]
- Calculation:
[tex]\[
n \hat{p} = 50 \times 0.9 = 45.0
\][/tex]
2. Compute [tex]\( n(1 - \hat{p}) \)[/tex]:
- Calculate [tex]\( 1 - \hat{p} \)[/tex]:
[tex]\[
1 - \hat{p} = 1 - 0.9 = 0.1
\][/tex]
- Then calculate:
[tex]\[
n(1 - \hat{p}) = 50 \times 0.1 = 5.0
\][/tex]
3. Check the conditions:
- First Condition: [tex]\( n \hat{p} \geq 10 \)[/tex]
- Calculated [tex]\( n \hat{p} = 45.0 \)[/tex], which is greater than 10. So, this condition is met.
- Second Condition: [tex]\( n(1 - \hat{p}) \geq 10 \)[/tex]
- Calculated [tex]\( n(1 - \hat{p}) = 5.0 \)[/tex], which is not greater than 10. So, this condition is not met.
Since both conditions are required to be at least 10 for the large counts condition to be satisfied, and we found that [tex]\( n(1 - \hat{p}) \)[/tex] is less than 10, 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. [tex]\( n \hat{p} \geq 10 \)[/tex]
2. [tex]\( n(1 - \hat{p}) \geq 10 \)[/tex]
Here's how you can solve this:
1. Compute [tex]\( n \hat{p} \)[/tex]:
- Given:
- Sample size [tex]\( n = 50 \)[/tex]
- Sample proportion [tex]\( \hat{p} = 0.9 \)[/tex]
- Calculation:
[tex]\[
n \hat{p} = 50 \times 0.9 = 45.0
\][/tex]
2. Compute [tex]\( n(1 - \hat{p}) \)[/tex]:
- Calculate [tex]\( 1 - \hat{p} \)[/tex]:
[tex]\[
1 - \hat{p} = 1 - 0.9 = 0.1
\][/tex]
- Then calculate:
[tex]\[
n(1 - \hat{p}) = 50 \times 0.1 = 5.0
\][/tex]
3. Check the conditions:
- First Condition: [tex]\( n \hat{p} \geq 10 \)[/tex]
- Calculated [tex]\( n \hat{p} = 45.0 \)[/tex], which is greater than 10. So, this condition is met.
- Second Condition: [tex]\( n(1 - \hat{p}) \geq 10 \)[/tex]
- Calculated [tex]\( n(1 - \hat{p}) = 5.0 \)[/tex], which is not greater than 10. So, this condition is not met.
Since both conditions are required to be at least 10 for the large counts condition to be satisfied, and we found that [tex]\( n(1 - \hat{p}) \)[/tex] is less than 10, 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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