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Chapter 9: Problem 45

For each of the following choices, explain which would result in a narrower large-sample confidence interval for \(p\) : a. \(95 \%\) confidence level or \(99 \%\) confidence level b. \(n=200\) or \(n=500\)

Short Answer

Expert verified
a) The \(95 \%\) confidence level will result in a narrower large-sample confidence interval for \(p\). b) A sample size of n=500 will result in a narrower large-sample confidence interval for \(p\).

Step by step solution

01

Understanding confidence intervals

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. The width of a confidence interval depends on the chosen confidence level and the sample variance. A higher confidence level will result in a wider confidence interval.
02

Analyzing the confidence levels

Given the confidence levels of \(95 \%\) or \(99 \%\) in scenario a), it can be understood that the \(99 \%\) confidence level will provide a wider range as compared to the \(95 \%\) confidence level. This is because a \(99 \%\) confidence level will be more precise and therefore will provide a larger interval range. So, the \(95 \%\) confidence level will result in a narrower large-sample confidence interval for \(p\).
03

Understanding the Sample Size

In statistics, when sample size increases, it provides more accurate and reliable results. It reduces the margin of error and narrows the confidence interval. This is all because increasing sample size tends to lead to a more precise estimate of the population parameter.
04

Analyzing the Sample Sizes

Given the two sample sizes of n=200 or n=500 in scenario b), it is clear that a larger sample size will provide more accurate results and hence a narrower confidence interval. So, in this case, n=500 will result in a narrower large-sample confidence interval for \(p\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Level
In statistics, the confidence level indicates the degree of certainty that the true parameter lies within the estimated range (or confidence interval). The confidence level is one of the key factors that determine how wide or narrow a confidence interval will be.
  • A higher confidence level, such as 99%, means that we are more certain the interval contains the true population parameter. This adds to the assurance but also results in a wider interval.
  • Conversely, a lower confidence level, like 95%, offers less certainty but results in a narrower interval. Therefore, there is a trade-off between confidence and precision.
By choosing a 95% confidence level instead of a 99% one, you accept a slightly lower level of certainty, but you benefit from a tighter, more precise estimate of your parameter.
Sample Size
Sample size is another important aspect that influences the confidence interval. Essentially, it reflects the number of observations or data points collected for a study.
  • An increase in sample size generally leads to more reliable and accurate estimates of the population parameter.
  • With a larger sample size, the sample becomes more representative of the entire population, thereby reducing the variability or the spread of the confidence interval.
In the context of choosing between a sample size of 200 or 500, a sample size of 500 would yield a narrower confidence interval because it provides more data for computing the estimate, hence increasing accuracy.
Margin of Error
The margin of error is a measure that tells us how much we can expect our sample statistic to fluctuate if we were to repeat the sample collection process multiple times.
  • It quantifies the level of uncertainty, and is affected by both the sample size and the confidence level.
  • The larger the sample size or the lower the confidence level, the smaller the margin of error will be, resulting in a narrower confidence interval.
This means that by opting for a bigger sample size or a less conservative confidence level, you can effectively reduce the margin of error, leading to a more concise estimate of the population parameter.

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Most popular questions from this chapter

For estimating a population characteristic, why is an unbiased statistic with a small standard error preferred over an unbiased statistic with a larger standard error?

Thereport"2005 ElectronicMonitoring\& Surveillance \(\begin{array}{lll}\text { Survey: } & \text { Many Companies Monitoring, } & \text { Recording, }\end{array}\) Videotaping-and Firing-Employees" (American Management nesses. The report stated that 137 of the 526 businesses had fired workers for misuse of the Internet. Assume that this sample is representative of businesses in the United States. a. Estimate the proportion of all businesses in the U.S. that have fired workers for misuse of the Internet. What statistic did you use? b. Use the sample data to estimate the standard error of \(\hat{p}\). c. Calculate and interpret the margin of error associated with the estimate in Part (a). (Hint: See Example 9.3 )

One thousand randomly selected adult Americans participated in a survey conducted by the Associated Press (June, 2006). When asked "Do you think it is sometimes justified to lie, or do you think lying is never justified?" \(52 \%\) responded that lying was never justified. When asked about lying to avoid hurting someone's feelings, 650 responded that this was often or sometimes OK. a. Construct and interpret a \(90 \%\) confidence interval for the proportion of adult Americans who would say that lying is never justified. b. Construct and interpret a \(90 \%\) confidence interval for the proportion of adult Americans who think that it is often or sometimes OK to lie to avoid hurting someone's feelings. c. Using the confidence intervals from Parts (a) and (b), comment on the apparent inconsistency in the responses.

In response to budget cuts, county officials are interested in learning about the proportion of county residents who favor closure of a county park rather than closure of a county library. In a random sample of 500 county residents, 198 favored closure of a county park. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1: It is unlikely that the estimate \(\hat{p}=0.396\) differs from the value of the actual population proportion by more than 0.0429 Statement 2: The estimate \(\hat{p}=0.396\) will never differ from the value of the actual population proportion by more than 0.0429 Statement 3: It is unlikely that the estimate \(\hat{p}=0.396\) differs from the value of the actual population proportion by more than 0.0219

The article "Career Expert Provides DOs and DON'Ts for Job Seekers on Social Networking" (CareerBuilder.com, August 19,2009 ) included data from a survey of 2,667 hiring managers and human resource professionals. The article noted that more employers are now using social networks to screen job applicants. Of the 2,667 people who participated in the survey, 1,200 indicated that they use social networking sites such as Facebook, MySpace, and LinkedIn to research job applicants. Assume that the sample is representative of hiring managers and human resource professionals. Answer the four key questions (QSTN) to confirm that the suggested method in this situation is a confidence interval for a population proportion.
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