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Thirty small communities in Connecticut (population near 10,000 each) gave an average of \(\bar{x}=138.5\) reported cases of larceny per year. Assume that \(\sigma\) is known to be 42.6 cases per year (Reference: Crime in the United States, Federal Bureau of Investigation). (a) Find a \(90 \%\) confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (b) Find a \(95 \%\) confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (c) Find a \(99 \%\) confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase? (c) Critical Thinking Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

Step by step solution

01

Understand Confidence Interval and Margin of Error

A confidence interval provides a range within which we expect the population mean to lie, based on a sample mean. The margin of error is half the width of this confidence interval. The formula for the confidence interval is given by \( \bar{x} \pm Z \frac{\sigma}{\sqrt{n}} \), where \( Z \) is the Z-score corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

02

Calculate Confidence Interval for 90% Confidence Level

For a 90% confidence level, the Z-score is approximately 1.645. Using \( \bar{x} = 138.5 \), \( \sigma = 42.6 \), and \( n = 30 \), calculate the margin of error:\[ME_{90} = Z \frac{\sigma}{\sqrt{n}} = 1.645 \times \frac{42.6}{\sqrt{30}} \approx 12.8\]Thus, the 90% confidence interval is:\[ (138.5 - 12.8, 138.5 + 12.8) = (125.7, 151.3) \]

03

Calculate Confidence Interval for 95% Confidence Level

For a 95% confidence level, the Z-score is approximately 1.96. The margin of error is:\[ME_{95} = 1.96 \times \frac{42.6}{\sqrt{30}} \approx 15.3\]Thus, the 95% confidence interval is:\[ (138.5 - 15.3, 138.5 + 15.3) = (123.2, 153.8) \]

04

Calculate Confidence Interval for 99% Confidence Level

For a 99% confidence level, the Z-score is approximately 2.576. The margin of error is:\[ME_{99} = 2.576 \times \frac{42.6}{\sqrt{30}} \approx 20.0\]Thus, the 99% confidence interval is:\[ (138.5 - 20.0, 138.5 + 20.0) = (118.5, 158.5) \]

05

Compare Margins of Error

The margins of error for 90%, 95%, and 99% confidence levels are 12.8, 15.3, and 20.0, respectively. As the confidence level increases, the margin of error also increases.

06

Compare Lengths of Confidence Intervals

The lengths of the confidence intervals for 90%, 95%, and 99% confidence levels are 25.6, 30.6, and 40, respectively. As the confidence level increases, the confidence intervals increase in length, reflecting the increase in margins of error.

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Key Concepts

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

Margin of Error

The margin of error is a crucial concept when working with confidence intervals. It represents the extent to which the estimate might differ from the true population parameter. In simpler terms, it's the buffer zone we add and subtract from our sample statistics to capture the population parameter with a certain level of confidence. For example, if your sample mean is 138.5 and the margin of error is 15.3, this means that the true population mean is likely between 123.2 and 153.8 when using a 95% confidence interval.

To calculate the margin of error, you'll use the formula:

• Margin of Error = \( Z \frac{\sigma}{\sqrt{n}} \)

where:

• \( Z \) is the Z-score associated with your chosen confidence level,
• \( \sigma \) is the population standard deviation,
• \( n \) is the sample size.

As the confidence level increases, the Z-score and thus the margin of error also increase. This results in a wider confidence interval, giving us more assurance that the true mean lies within that range.

Z-score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. Essentially, it tells us how many standard deviations away a particular value is from the mean.

When calculating confidence intervals, the Z-score corresponds to the desired confidence level. For example:

• A 90% confidence level has a Z-score of approximately 1.645.
• A 95% confidence level has a Z-score of approximately 1.96.
• A 99% confidence level has a Z-score of approximately 2.576.

This score is crucial because it influences the margin of error directly. As the confidence level increases, the Z-score increases, leading to a larger margin of error. This means the interval needs to be wider to ensure that it captures the true population mean with higher confidence.

Population Standard Deviation

The population standard deviation, denoted by \( \sigma \), is a measure of the spread or dispersion of a set of values in a population. It tells us how much individual values differ from the average value of the dataset.

In the context of confidence intervals, the population standard deviation helps determine the margin of error. It does so by scaling the impact of the Z-score. The formula for the margin of error is:

• \( Z \frac{\sigma}{\sqrt{n}} \)

The population standard deviation plays a pivotal role as it reflects the variability in the data. The larger the standard deviation, the wider the margin of error, leading to a broader confidence interval. This indicates there is more variability in the data, and consequently, more uncertainty about where the true population mean lies.

Sample Size

The sample size, denoted by \( n \), refers to the number of observations or data points collected from the population. It is a key factor in determining the reliability and precision of statistical estimates.

In the formation of confidence intervals, the sample size affects the margin of error inversely. The larger the sample size, the smaller the margin of error, and the more precise the confidence interval becomes. This relationship can be seen in the formula for the margin of error:

• \( Z \frac{\sigma}{\sqrt{n}} \)

Whereas a larger sample size effectively reduces the standard error \( (\frac{\sigma}{\sqrt{n}}) \), it results in a smaller margin of error. Thus, to achieve a narrower and more dependable confidence interval, increasing the sample size is often a practical step.