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Chapter 9: Problem 76

In each of the following cases, do you think the sample size is large enough to use the normal distribution to make a test of hypothesis about the population proportion? Explain why or why not. a. \(n=30\) and \(p=.65\) b. \(n=70\) and \(p=.05\) c. \(n=60\) and \(p=.06\) d. \(n=900\) and \(p=.17\)

Short Answer

Expert verified
Normal distribution can be applied for cases a and d, not for cases b and c.

Step by step solution

01

Calculate np and n(1-p) for case a

We start with case a. The given values are \(n=30\) and \(p=.65\). Calculate \(np = 30 * 0.65 = 19.5\) and \(n(1-p) = 30 * (1-0.65) = 10.5\).
02

Assess Normal Approximation for case a

Since both \(np = 19.5\) and \(n(1-p) = 10.5\) are greater than 5, it is safe to apply the normal approximation for case a.
03

Repeat Steps 1 and 2 for other cases

Repeat the process followed in Steps 1 and 2 for the other cases. For case b, we have \(np = 70*0.05 = 3.5, n(1-p)=66.5\). For case c, we have \(np = 60*0.06 = 3.6, n(1-p)=56.4\). And for case d, \(np = 900*0.17 = 153, n(1-p)=747\).
04

Conclude

Based on the computations, the normal approximation will not be applicable for cases b and c, as their respective \(np\) values are less than 5. However, it will be safe to apply for cases a and d as both their \(np\) and \(n(1-p)\) values are greater than 5.

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Most popular questions from this chapter

Records in a three-county area show that in the last few years, Girl Scouts sell an average of \(47.93\) boxes of cookies per year, with a population standard deviation of \(8.45\) boxes per year. Fifty randomly selected Girl Scouts from the region sold an average of \(46.54\) boxes this year. Scout leaders are concerned that the demand for Girl Scout cookies may have decreased. a. Test at the \(10 \%\) significance level whether the average number of boxes of cookies sold by all Girl Scouts in the three-county area is lower than the historical average. b. What will your decision be in part a if the probability of a Type I error is zero? Explain.

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