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Chapter 7: Problem 55

Indicate in which of the following cases the central limit theorem will apply to describe the sampling distribution of the sample proportion. a. \(n=20\) and \(p=.45\) b. \(n=75\) and \(p=.22\) c. \(n=350\) and \(p=.01\) d. \(n=200\) and \(p=.022\)

Short Answer

Expert verified
The Central Limit Theorem can be applied in case b only.

Step by step solution

01

Calculate np and n(1-p) in each case

For case a (n=20, p=.45), np=9 and n(1-p)=11. For case b (n=75, p=.22), np=16.5 and n(1-p)=58.5. For case c (n=350, p=.01), np=3.5 and n(1-p)=346.5. For case d (n=200, p=.022), np=4.4 and n(1-p)=195.6.
02

Determine whether both np and n(1-p) are greater than or equal to 10 in each case

For case a, np is less than 10, so the central limit theorem does not apply. For case b, both np and n(1-p) are greater than 10, so the central limit theorem applies. For case c, np is less than 10, so the central limit theorem does not apply. For case d, np is less than 10, so the central limit theorem does not apply.
03

Summarize the results

The Central Limit Theorem applies to the sampling distribution of the sample proportion only in case b, where both np and n(1-p) are greater than or equal to 10. In the other cases, the Central Limit Theorem does not apply because np is less than 10.

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