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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? (e) 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 Intervals

A confidence interval estimates a range for a population parameter based on sample data. It is calculated using the formula \( \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \), where \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the confidence level, \(\sigma\) is the known population standard deviation, and \(n\) is the sample size.

02

Calculate 90% Confidence Interval

For a 90% confidence interval, the critical z-score is approximately 1.645. Using \( \bar{x} = 138.5 \), \( \sigma = 42.6 \), and \( n = 30 \), calculate the margin of error as 1.645 \( \times \frac{42.6}{\sqrt{30}} \approx 12.78 \). The confidence interval is then \( 138.5 \pm 12.78 \), or (125.72, 151.28).

03

Calculate 95% Confidence Interval

For a 95% confidence interval, the z-score is approximately 1.96. The margin of error is 1.96 \( \times \frac{42.6}{\sqrt{30}} \approx 15.24 \). Thus, the confidence interval is \( 138.5 \pm 15.24 \), or (123.26, 153.74).

04

Calculate 99% Confidence Interval

For a 99% confidence interval, the critical z-score is about 2.576. This gives a margin of error of 2.576 \( \times \frac{42.6}{\sqrt{30}} \approx 20.04 \). The confidence interval is \( 138.5 \pm 20.04 \), or (118.46, 158.54).

05

Compare Margins of Error

The margins of error for the confidence intervals are 12.78 for 90%, 15.24 for 95%, and 20.04 for 99%. As the confidence level increases, the margin of error increases. Indeed, higher confidence requires accounting for more variability, thus a larger range.

06

Compare Confidence Interval Lengths

The lengths of the intervals are 25.56 for 90%, 30.48 for 95%, and 40.08 for 99%. As with margins of error, higher confidence levels result in longer confidence intervals.

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Key Concepts

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

Understanding Margin of Error

The margin of error is an important concept in statistics that helps us understand the precision of our estimate concerning a population parameter. It tells us how much the sample result could vary due to random sampling variations.

The formula for calculating the margin of error is: - Margin of Error = z-score \( \times \) \( \frac{\sigma}{\sqrt{n}} \) Where:

• z-score relates to the confidence level chosen.
• \(\sigma\) is the population standard deviation.
• \(n\) is the sample size.

The margin of error gives us half the width of a confidence interval. For example, a margin of error of 12.78 means that the true population mean could be 12.78 units above or below our sample mean.

Population Parameter Estimation

When we talk about estimating a population parameter, we're often aiming to make conclusions about a large group based on data from a smaller group, or sample. In this scenario, the sample is from various communities reporting larceny cases.

Estimating a single "true" value directly from a sample can often be challenging due to variability in the data. Instead, a confidence interval provides a range of values within which the true population parameter is likely to lie.

• A 90% confidence interval means there's a 90% chance that the interval contains the population mean.
• Similar logic applies to 95% and 99% confidence intervals, with increasing levels of certainty.

This method represents an interval estimate rather than a point estimate, allowing us to account for randomness in the data collection process.

Explaining Z-score

The z-score is a statistical measure that describes a value's position relative to the mean of a group of values. It's expressed in terms of standard deviations.

In the context of confidence intervals, the z-score determines how many standard deviations away we want to reach to enclose a certain percentage of the normal distribution.

• For a 90% confidence interval, the z-score is typically 1.645.
• For a 95% confidence interval, it is usually 1.96.
• For a 99% confidence interval, the z-score is approximately 2.576.

These z-scores ensure the desired percentage of the population distribution is captured between the confidence limits.

Understanding Confidence Levels

A confidence level refers to the degree of certainty that the true parameter lies within the confidence interval. This confidence is expressed as a percentage and decides the width of the interval. Essentially, a higher confidence level gives more assurance but results in a broader range.

For example:

• A 90% confidence level indicates a smaller range, meaning there's less certainty but the intervals are narrower.
• Conversely, a 99% confidence level aims for higher certainty, making the intervals wider.

The balance between confidence level and interval width is crucial. Higher confidence levels require more variability coverage, thus enforcing longer intervals. Therefore, choosing the appropriate confidence level involves weighing the risk of error against practicability and maintaining a useful level of precision.