Question

Have You Ever Been Arrested? According to a recent study of 7335 young people in the US, \(30 \%\) had been arrested \(^{28}\) for a crime other than a traffic violation by the age of 23. Crimes included such things as vandalism, underage drinking, drunken driving, shoplifting, and drug possession. (a) Is the \(30 \%\) a parameter or a statistic? Use the correct notation. (b) Use the information given to estimate a parameter, and clearly define the parameter being estimated. (c) The margin of error for the estimate in part (b) is \(0.01 .\) Use this information to give a range of plausible values for the parameter. (d) Given the margin of error in part (c), if we asked all young people in the US if they have ever been arrested, is it likely that the actual proportion is less than \(25 \% ?\)

Short answer

(a) The 30% is a statistic. (b) The parameter being estimated is the actual percentage of young people in the U.S who have been arrested for a crime other than a traffic violation by the age of 23, and it is estimated as around 30% (c) The range of plausible values for the parameter is between 29% and 31% (d) No, it is not likely that the actual proportion is less than 25%

Step-by-step solution

Identify the Parameter or Statistic

A parameter is a value that describes a characteristic of a population, and a statistic is a value that describes a characteristic of a sample. The fact reveal that \(30 \%\) of 7335 young people had been arrested for a crime other than a traffic violation by the age of 23. Since this information is gathered from a sample (7335 young people), the \(30 \%\) is a statistic.

Estimating the Parameter

Since the \(30 \%\) is derived from a sample, it can be used to estimate the parameter which is the actual percentage of young people in the U.S who have been arrested for a crime other than a traffic violation by the age of 23. Let's denote this parameter as \(p\). So we can estimate, \(p \approx 30 \% \)

Calculating the Range of Plausible Values

The question provides a margin of error of \(0.01\) for this estimate. This means that the actual value of the parameter can be \(0.01\) above or below the estimated value. Therefore, the range of plausible values for the parameter \(p\) is \(p \in [30\%-0.01 , 30\%+0.01] = [29\%, 31\%] \). So, the percentage of young people in the U.S who have been arrested for a crime other than a traffic violation by the age of 23 could plausibly be between 29% and 31%.

Evaluate the Possibility of the Proportion Being Less Than 25%

Given the calculated range of plausible values for parameter \(p\) (29% to 31%), it is unlikely that the actual proportion of young people in the U.S who have been arrested for a crime other than a traffic violation by the age of 23 is less than 25%.

Key concepts

Parameter vs Statistic

Understanding the difference between a parameter and a statistic is critical for students diving into the world of statistical inference. In the exercise, we are introduced to a study involving a sample of 7,335 young people in the United States. A 'parameter' is a numerical characteristic or measure of an entire population, whereas a 'statistic' is the equivalent measure but derived from a sample of the population. In this case, the reported 30% is a statistic because it reflects the proportion of individuals within the sampled group of 7,335 who have been arrested, rather than the entire population of young people in the US.

Correct notation plays a significant role in statistics, so we denote population parameters typically with Greek letters (like \(\mu\) for the population mean or \(p\) for a population proportion) and statistics with Latin letters (like \(\bar{x}\) for the sample mean or \(\hat{p}\) for the sample proportion).

Margin of Error

The 'margin of error' is a term that can puzzle many students, but it's essential for interpreting survey results. It quantifies the uncertainty in estimating a population parameter. This measure gives us a range within which the true parameter value is likely to fall. For instance, in the given exercise, a margin of error of 0.01 means we can be confident that the actual population proportion (parameter) of young people who have been arrested for non-traffic crimes is within 1% of our observed statistic of 30%.

This concept allows researchers and the public alike to grasp the precision of the statistical estimate. With a margin of error of 0.01, we infer that the true proportion lies in the interval \[29\%, 31%\]. So, it's not just about the point estimate (the 30%); the margin of error provides the context needed to understand the estimate's reliability and to make informed decisions based on that data.

Estimating Parameters

When we talk about 'estimating parameters,' we refer to the process of using statistics calculated from a sample to infer or predict the value of a corresponding parameter in the population. The study under discussion has led to the estimation that approximately 30% of all young people in the US could have been arrested for non-traffic related crimes by the age of 23.

However, to fully understand this estimate's implications, we must not only calculate it correctly, often using a formula or a model, but also assess its precision and possible error range. In practice, statisticians use confidence intervals and hypothesis tests to make judgements about parameters based on sample statistics. These procedures account for sampling variability and provide a structured approach to estimation that aids in making sound statistical inferences. The improvement in understanding can come from visualizing this concept, maybe by plotting confidence intervals on a graph, or through simulation that demonstrates how sample statistics may vary and yet provide good estimates for the population parameters.