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Multiple choice: Select the best answer for Exercises 49 to 52. I collect an SRS of size n from a population and compute a 95% confidence interval for the population proportion. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data? (a) Use a larger confidence level. (b) Use a smaller confidence level. (c) Increase the sample size. (d) Use the same confidence level, but compute the interval n times. Approximately 5% of these intervals will be larger. (e) Nothing can guarantee absolutely that you will get a larger interval. One can only say that the chance of obtaining a larger interval is 0.05.

Step by step solution

01

Understand the Concept of Confidence Intervals

A confidence interval provides a range of values that is believed to contain the true population parameter (in this case, the population proportion) with a certain level of confidence. The width of the interval is influenced by the confidence level and the sample size.

02

Consider the Effect of Confidence Level on Interval Width

When the confidence level increases (from 95% to 99%, for example), the interval becomes wider because you need more leeway in your estimates to ensure a higher degree of confidence. Thus, using a larger confidence level results in a wider confidence interval.

03

Analyze the Impact of Sample Size

Increasing the sample size typically decreases the width of the confidence interval because the estimate becomes more precise. This is contrary to the requirement of obtaining a larger interval.

04

Evaluate Each Option

- **Option (a)** suggests using a larger confidence level, which increases interval width; therefore, it answers the question correctly. - **Option (b)** involves a smaller confidence level, leading to a narrower interval. - **Option (c)**, increasing sample size, results in a narrower interval. - **Option (d)** does not consistently produce larger intervals, as it focuses on repeated sampling leading to variations. - **Option (e)** suggests that no absolute guarantee can be given, which is true but not the best choice for producing a larger interval.

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Key Concepts

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

Confidence Level

The confidence level is a key factor in determining the width of a confidence interval. It represents the degree of certainty that the population parameter lies within the interval. Common confidence levels include 90%, 95%, and 99%.

• A higher confidence level, such as 99%, means you want to be more certain that the interval contains the true parameter. To achieve this, the interval must be wider, providing more room to cover potential variability.
• Conversely, a lower confidence level, like 90%, means you accept more risk of the interval missing the true parameter, resulting in a narrower interval.

By choosing a higher confidence level, you increase the likelihood that you capture the true population parameter. However, this also means a larger margin of error, thus a wider interval. This is essential when precision in estimates is less important than certainty of coverage. Understanding this trade-off is crucial for interpreting confidence intervals effectively.

Sample Size

Sample size greatly influences the precision of a confidence interval. It refers to the number of observations or data points in a sample. A key principle in statistics is that larger samples tend to provide more reliable estimates.

• Increasing the sample size usually results in a narrower confidence interval. This is because more data points offer a better approximation of the population parameter, reducing the margin of error.
• On the contrary, a smaller sample size yields a wider interval since there is more uncertainty and variability in the estimates.

While larger samples can reduce uncertainty, they also require more resources and time to collect. Deciding on an appropriate sample size depends on the balance between desired precision and available resources. In general, if a narrower interval and more precise results are desired, increasing the sample size is an effective strategy.

Margin of Error

The margin of error quantifies the range of uncertainty in a confidence interval. It is the extent the sample proportion differs from the true population proportion within the defined confidence level. Several factors influence the margin of error:

• A higher confidence level increases the margin of error, leading to a wider interval, as more uncertainty is accounted for to increase the confidence that the interval captures the true parameter.
• A larger sample size decreases the margin of error since more data points provide more precise estimates, tightening the interval.

Reducing the margin of error is essential for accurate estimates. However, this reduction must be balanced with the level of confidence desired, as obtaining both a small margin of error and a high confidence level can be resource-intensive. Understanding the margin of error helps in evaluating how much uncertainty is present and how it impacts the overall confidence interval.