384 Chapter 8 Confidence Intervals Check Your Understanding 4. A pollster is planning to draw a simple random sample of 500 people in Colorado (population 5.2 million). He then will conduct a similar poll in Texas (population 26.1 million). He wants to have approximately the same standard error in both polls. True or false: a. The pollster needs a sample in Texas that is about 5 times as large as the one in Colorado. False b. The pollster needs a sample in Texas that is about the same size as the one in Colorado. True 5. A marketing firm in New York City (population 8.3 million) plans to draw a simple random sample of 1000 people to estimate the proportion who have heard about a new product. The firm then plans to take a simple random sample of 500 in Denver (population 634,000) for the same purpose. True or false: a. The margin of error for a 95% confidence interval will be larger in New York. False b. The margin of error for a 95% confidence interval will be larger in Denver. True c. The margin of error for a 95% confidence interval will be about the same in both cities. False Answers are on page 390. Objective 3 Describe a method for constructing confidence intervals with small samples A Method for Constructing Confidence Intervals with Small Samples The method that we have presented for constructing a confidence interval for a proportion requires that we have at least 10 individuals in each category. When this condition is not met, we can still construct a confidence interval by adjusting the sample proportion a bit. We increase the number of individuals in each category by 2, so that the sample size increases by 4. Thus, instead of using the sample proportion ̂p = x∕n, we use the adjusted sample proportion ̃p = x + 2 n + 4 The standard error and critical value are calculated in the same way as in the traditional method, except that we use the adjusted sample proportion ̃p in place of ̂p, and n + 4 in place of n. Constructing Confidence Intervals for a Proportion with Small Samples If x is the number of individuals in a sample of size n who have a certain characteristic, and p is the population proportion, then: The adjusted sample proportion is ̃p = x + 2 n + 4 A confidence interval for p is ̃p − z��∕2 √ ̃p(1 − ̃p) n + 4 < p < ̃p + z��∕2 √ ̃p(1 − ̃p) n + 4 Another way to write this is ̃p ± z��∕2 √ ̃p(1 − ̃p) n + 4
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