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FBI Report: Larceny 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? (e) 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

Understanding Confidence Interval Calculation

To calculate the confidence interval, use the formula: \[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \] where \( \bar{x} = 138.5 \), \( \sigma = 42.6 \), \( n = 30 \), and \( z \) is the z-score corresponding to the confidence level.

02

Calculate the 90% Confidence Interval

For a 90% confidence interval, the z-score is approximately 1.645. Plug in the values:\[ 138.5 \pm 1.645 \frac{42.6}{\sqrt{30}} \]Calculate the margin of error:\[ 1.645 \times \frac{42.6}{\sqrt{30}} \approx 12.78 \]Thus, the confidence interval is:\[ (125.72, 151.28) \]

03

Calculate the 95% Confidence Interval

For a 95% confidence interval, the z-score is approximately 1.96. Plug in the values:\[ 138.5 \pm 1.96 \frac{42.6}{\sqrt{30}} \]Calculate the margin of error:\[ 1.96 \times \frac{42.6}{\sqrt{30}} \approx 15.26 \]Thus, the confidence interval is:\[ (123.24, 153.76) \]

04

Calculate the 99% Confidence Interval

For a 99% confidence interval, the z-score is approximately 2.576. Plug in the values:\[ 138.5 \pm 2.576 \frac{42.6}{\sqrt{30}} \]Calculate the margin of error:\[ 2.576 \times \frac{42.6}{\sqrt{30}} \approx 20.04 \]Thus, the confidence interval is:\[ (118.46, 158.54) \]

05

Compare Margins of Error

Compare the margins of error: for 90% it's 12.78, for 95% it's 15.26, and for 99% it's 20.04. As the confidence level increases, the margin of error also increases.

06

Compare Lengths of Confidence Intervals

Compare the lengths of the confidence intervals: the length for 90% is 25.56, for 95% is 30.52, and for 99% is 40.08. As the confidence level increases, the length of the confidence interval increases.

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Key Concepts

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

Margin of Error

In statistics, the margin of error is a crucial concept when discussing confidence intervals. It's essentially the buffer zone around a sample mean within which we expect the true population mean to lie. This value gives us an idea of how much "wiggle room" we have around our estimate. The formula to find the margin of error is:

• First, determine the z-score, which corresponds to your desired confidence level.
• Next, multiply this z-score by the standard deviation (\( \sigma \)) divided by the square root of the sample size (n). This is the formula: \( z \times \frac{\sigma}{\sqrt{n}} \).

It is important to note that as you increase the confidence level, the z-score gets larger, making the margin of error wider. This reflects your need for more certainty in your interval, which requires "stretching" the margin.

Z-Score

The z-score is a central element in determining confidence intervals. It represents how many standard deviations away a particular data point is from the mean. When calculating confidence intervals, you select a z-value corresponding to how confident you want to be:

• Common confidence levels include 90%, 95%, and 99%.
• Each level has a specific z-score, which can be found in a z-table or through statistical software.
• For example, a 90% confidence interval uses a z-score of approximately 1.645, 95% uses 1.96, and 99% uses 2.576.

These scores ensure that the percentage of the distribution under the normal curve gives you the desired confidence. Essentially, the higher the confidence level, the higher the z-score, leading to a wider confidence interval.

Statistical Analysis

Statistical analysis is the umbrella under which confidence intervals, margin of error, and z-scores fall. It entails collecting and scrutinizing data to identify patterns and make inferences:

• The primary goal is to make informed decisions and predictions about a population based on sample data.
• By employing confidence intervals, you can estimate population parameters with known levels of reliability.
• Z-scores assist in determining how data points relate to the mean, allowing comparison across different datasets.
• Margins of error provide a range that accounts for sampling variability, creating a more comprehensive view of potential outcomes.

By utilizing these statistical techniques, individuals can move beyond merely observing data to truly understanding and interpreting their significance.