A 2003 survey showed that 4.8 percent of the 250 Americans...

Name: Problem 31, Chapter 8: Applied Statistics in Business and Economics
Uploaded: 2023-07-23T13:14:03-08:00
Duration: 2 min 2 s
Channel: Sheryl Ezze
Description: A 2003 survey showed that 4.8 percent of the 250 Americans surveyed had suffered some kind of identity theft in the past 12 months. (a) Construct a 99 percent confidence interval for the true proportion of Americans who had suffered identity theft in the past 12 months. (b) May normality be assumed? Explain.

A 2003 survey showed that 4.8 percent of the 250 Americans surveyed had suffered some kind of identity theft in the past 12 months. (a) Construct a 99 percent confidence interval for the true proportion of Americans who had suffered identity theft in the past 12 months. (b) May normality be assumed? Explain.

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Key Concepts

Standard Error

The standard error measures the variability of the sample proportion as an estimate of the true population proportion. It is calculated from the sample proportion and the sample size, and plays a critical role in determining the width of the confidence interval.

Normal Approximation to the Binomial

Normal approximation to the binomial distribution is justified when the sample size is large enough such that both np and n(1-p) are greater than or equal to a threshold (commonly 10). This allows the use of the central limit theorem to assume that the sampling distribution of the proportion is approximately normal, facilitating easier calculation of confidence intervals and hypothesis tests.

Confidence Interval for a Proportion

A confidence interval for a proportion provides a range of values that is likely to contain the true population proportion with a specified level of confidence, which is determined from sample data and the estimated standard error. This interval helps gauge the uncertainty associated with the sample estimate.

Sample Proportion

The sample proportion is the ratio of individuals in the sample exhibiting a particular characteristic to the total number of individuals sampled. This statistic serves as an estimate for the true population proportion and is fundamental in constructing interval estimates.

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