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Chapter 8: Problem 51

What assumption(s) must hold true to use the normal distribution to make a confidence interval for the population proportion, \(p\) ?

Short Answer

Expert verified
The assumptions required are: the sample must be random, the sampling distribution should be approximately normally distributed (usually possible if sample size is large enough, n > 30), and the sampled observations must be independent (practically, if sample size is less than 10% of the population, independence is safe to assume).

Step by step solution

01

Identify the Assumptions to Use the Normal Distribution

There are several assumptions that need to be satisfied to use the normal distribution to make a confidence interval for the population proportion:
02

Assumption 1: Randomness

The first assumption is that the sample must be random; that is, each individual in the population has an equal chance of being selected.
03

Assumption 2: Normal Distribution

Secondly, the sampling distribution should be approximately normally distributed. This is typically possible if the sample size is large enough, usually n > 30, according to the Central Limit Theorem.
04

Assumption 3: Independence

The third assumption is independence. This means that the probability that one individual is included in the sample does not affect the probability that any other individual is included. In more practical terms, it's pretty safe to assume independence if your sample size is less than 10% of the population size.

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Most popular questions from this chapter

A large city with chronic economic problems is considering legalizing casino gambling. The city council wants to estimate the proportion of all adults in the city who favor legalized casino gambling. What is the most conservative estimate of the minimum sample size that would limit the margin of error to be within \(.05\) of the population proportion for a \(95 \%\) confidence interval?

A city planner wants to estimate the average monthly residential water usage in the city at a \(97 \%\) confidence level. Based on earlier data, the population standard deviation of the monthly residential water usage in this city is \(389.60\) gallons. How large a sample should be selected so that the estimate for the average monthly residential water usage in this city is within 100 gallons of the population mean?

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