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Answer :
To determine if the large counts condition is met, we need to ensure that the values of [tex]\( n\hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] are both at least 10. Here's how you can verify these conditions:
1. Identify the values:
- Sample size [tex]\( n = 50 \)[/tex]
- Estimated proportion [tex]\( \hat{p} = 0.9 \)[/tex]
2. Calculate [tex]\( n\hat{p} \)[/tex]:
- This is the product of the sample size and the estimated proportion.
- [tex]\( n\hat{p} = 50 \times 0.9 = 45 \)[/tex]
3. Calculate [tex]\( n(1-\hat{p}) \)[/tex]:
- First, find [tex]\( 1-\hat{p} \)[/tex], which is [tex]\( 1 - 0.9 = 0.1 \)[/tex].
- Then, calculate the product of the sample size and this value.
- [tex]\( n(1-\hat{p}) = 50 \times 0.1 = 5 \)[/tex]
4. Check the large counts condition:
- For the condition to be met, both [tex]\( n\hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] should be at least 10.
- [tex]\( n\hat{p} = 45 \)[/tex] is at least 10, so this part of the condition is met.
- However, [tex]\( n(1-\hat{p}) = 5 \)[/tex] is not at least 10, so this part of the condition is not met.
Since [tex]\( n(1-\hat{p}) \)[/tex] is not at least 10, the large counts condition is not fully satisfied. Thus, the correct answer is:
No, [tex]\( n\hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] are not both at least 10.
1. Identify the values:
- Sample size [tex]\( n = 50 \)[/tex]
- Estimated proportion [tex]\( \hat{p} = 0.9 \)[/tex]
2. Calculate [tex]\( n\hat{p} \)[/tex]:
- This is the product of the sample size and the estimated proportion.
- [tex]\( n\hat{p} = 50 \times 0.9 = 45 \)[/tex]
3. Calculate [tex]\( n(1-\hat{p}) \)[/tex]:
- First, find [tex]\( 1-\hat{p} \)[/tex], which is [tex]\( 1 - 0.9 = 0.1 \)[/tex].
- Then, calculate the product of the sample size and this value.
- [tex]\( n(1-\hat{p}) = 50 \times 0.1 = 5 \)[/tex]
4. Check the large counts condition:
- For the condition to be met, both [tex]\( n\hat{p} \)[/tex] and [tex]\( n(1-\hat{p}) \)[/tex] should be at least 10.
- [tex]\( n\hat{p} = 45 \)[/tex] is at least 10, so this part of the condition is met.
- However, [tex]\( n(1-\hat{p}) = 5 \)[/tex] is not at least 10, so this part of the condition is not met.
Since [tex]\( n(1-\hat{p}) \)[/tex] is not at least 10, the large counts condition is not fully satisfied. Thus, 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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