QuestionJuly 23, 2025

For which of the following sample proportion distributions does the Central Limit Theorem apply? n=20hat (p)=0.3 n=20hat (p)=0.2 n=20hat (p)=0.8 n=20hat (p)=0.1

For which of the following sample proportion distributions does the Central Limit Theorem apply? n=20hat (p)=0.3 n=20hat (p)=0.2 n=20hat (p)=0.8 n=20hat (p)=0.1
For which of the following sample proportion distributions does the Central Limit Theorem apply? n=20hat (p)=0.3 n=20hat (p)=0.2 n=20hat (p)=0.8 n=20hat (p)=0.1

Solution
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Answer

n=20, \hat{p}=0.3 Explanation 1. Check Central Limit Theorem Condition The Central Limit Theorem applies if np \geq 5 and n(1-p) \geq 5. Calculate for each case. 2. Calculate for n=20, \hat{p}=0.3 np = 20 \times 0.3 = 6, n(1-p) = 20 \times 0.7 = 14. Both conditions are satisfied. 3. Calculate for n=20, \hat{p}=0.2 np = 20 \times 0.2 = 4, n(1-p) = 20 \times 0.8 = 16. First condition is not satisfied. 4. Calculate for n=20, \hat{p}=0.8 np = 20 \times 0.8 = 16, n(1-p) = 20 \times 0.2 = 4. Second condition is not satisfied. 5. Calculate for n=20, \hat{p}=0.1 np = 20 \times 0.1 = 2, n(1-p) = 20 \times 0.9 = 18. First condition is not satisfied.

Explanation

1. Check Central Limit Theorem Condition<br /> The Central Limit Theorem applies if $np \geq 5$ and $n(1-p) \geq 5$. Calculate for each case.<br /><br />2. Calculate for $n=20, \hat{p}=0.3$<br /> $np = 20 \times 0.3 = 6$, $n(1-p) = 20 \times 0.7 = 14$. Both conditions are satisfied.<br /><br />3. Calculate for $n=20, \hat{p}=0.2$<br /> $np = 20 \times 0.2 = 4$, $n(1-p) = 20 \times 0.8 = 16$. First condition is not satisfied.<br /><br />4. Calculate for $n=20, \hat{p}=0.8$<br /> $np = 20 \times 0.8 = 16$, $n(1-p) = 20 \times 0.2 = 4$. Second condition is not satisfied.<br /><br />5. Calculate for $n=20, \hat{p}=0.1$<br /> $np = 20 \times 0.1 = 2$, $n(1-p) = 20 \times 0.9 = 18$. First condition is not satisfied.
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