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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

In a January 2014 survey conducted by the Associated PressWe TV, \(68 \%\) of American adults said that owning a home is the most important thing or \(a\) very important but not the most important thing (opportunityagenda.org). Assume that this survey was based on a random sample of 900 American adults. a. Construct a \(95 \%\) confidence interval for the proportion of all American adults who will say that owning a home is the most important thing or a very important but not the most important thing. b. Explain why we need to construct a confidence interval. Why can we not simply say that \(68 \%\) of all American adults would say that owning a home is the most important thing or \(a\) very important but not the most important thing?

A random sample of 20 managers was taken, and they were asked whether or not they usually take work home. The responses of these managers are given below, where yes indicates they usually take work home and \(n o\) means they do not. \(\begin{array}{llllllllll}\text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { No } \\ \text { Yes } & \text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { No } & \text { Yes }\end{array}\) Make a \(99 \%\) confidence interval for the percentage of all managers who take work home.

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?

It is said that happy and healthy workers are efficient and productive. A company that manufactures exercising machines wanted to know the percentage of large companies that provide on-site health club facilities. A random sample of 240 such companies showed that 96 of them provide such facilities on site. a. What is the point estimate of the percentage of all such companies that provide such facilities on site? b. Construct a \(97 \%\) confidence interval for the percentage of all such companies that provide such facilities on site. What is the margin of error for this estimate?

The standard deviation for a population is \(\sigma=14.8\). A random sample of 25 observations selected from this population gave a mean equal to \(143.72\). The population is known to have a normal distribution. a. Make a \(99 \%\) confidence interval for \(\mu\). b. Construct a \(95 \%\) confidence interval for \(\mu\). c. Determine a \(90 \%\) confidence interval for \(\mu\). d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) decrease as the confidence level decreases? Explain your answer.
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