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Chapter 3: Problem 51

Have You Ever Been Arrested? According to a recent study of 7335 young people in the US, \(30 \%\) had been arrested \(^{21}\) 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

Expert verified
The '30%' is a statistic, not a parameter. The parameter is the proportion of young people in the US who have been arrested for non-traffic crimes before age 23, and it is estimated to be around '30%'. Given a margin of error of 0.01, the parameter is likely between 0.29 and 0.31. It is unlikely that the actual proportion is less than '25%'.

Step by step solution

01

Identifying whether '30%' is a parameter or a statistic

The '30%' is a statistic because it was obtained from the sample of 7335 young people, which is a subset of the entire population of young people in the US.
02

Estimating a parameter and defining it

The parameter being estimated is the proportion of young people in the entire US who have been arrested for non-traffic related crimes before they turned 23. This parameter can be denoted as \(P\), and the given statistic, '30%', can be used as an estimate of this parameter, that is, \(P = 0.3\).
03

Providing a range of plausible values for the parameter

The margin of error for this estimate is 0.01. The range of plausible values for the parameter \(P\) is therefore from \(P - E\) to \(P + E\), where \(E\) is the margin of error. Substituting the given values, we have \(P - E = 0.3 - 0.01 = 0.29\) and \(P + E = 0.3 + 0.01 = 0.31\). Thus, the range of plausible values for \(P\) is from 0.29 to 0.31.
04

Determining the possibility of the actual proportion being less than '25%'

Given the margin of error and the estimated parameter, the plausible range of values for the parameter is between 0.29 and 0.31. This range is greater than '25%', and so it is very unlikely that the actual proportion of young people in the US who have been arrested for non-traffic related crimes before they turned 23 is less than '25%'.

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Key Concepts

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

Parameter vs Statistic
Understanding the difference between a parameter and a statistic is crucial in the field of statistics. A parameter is a value that describes a characteristic of an entire population. Since it's not feasible to examine every member of a population, we often don't know the true parameter value. On the other hand, a statistic is a value that describes a characteristic of a sample, which is a subset of the population.

For example, when a study includes surveying a group of 7335 young people to assess the proportion who have been arrested for non-traffic crimes before age 23, the result of 30% is a statistic. It's based on the sample, not the entire population. In contrast, the actual proportion of all young people in the US who have been arrested by that age would represent a parameter, denoted by 'P' in statistical notation. It is this parameter that the study seeks to estimate through the statistic.
Margin of Error
The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It represents how much the estimate might differ from the true population parameter due to random chance. The smaller the margin of error, the closer we can expect the sample statistic to match the population parameter.

In our example, the margin of error is 0.01 or 1%. This means that the true population proportion of young people who have been arrested for non-traffic crimes by the age of 23 could reasonably be 1% higher or lower than the 30% statistic obtained from the sample. This measure of uncertainty is crucial when interpreting survey results, as it gives us a range of plausible values for the parameter.
Confidence Interval
A confidence interval gives us a range of values, derived from the sample statistic, that is likely to contain the population parameter. This interval is constructed using the margin of error and gives a scope within which the true parameter value is estimated to lie, with a certain level of confidence.

In the exercise, with a margin of error of 0.01, we create a confidence interval for the parameter of young people in the US having been arrested for non-traffic crimes by age 23. By subtracting and adding the margin of error to the statistic (30%), we attain a confidence interval of 29% to 31%. This means we are reasonably confident that the true parameter lies within this interval. Notably, since this interval does not include 25%, we infer that it is unlikely the actual proportion is less than 25%, based on our obtained data and the calculated confidence interval.

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Most popular questions from this chapter

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Give information about the proportion of a sample that agrees with a certain statement. Use StatKey or other technology to estimate the standard error from a bootstrap distribution generated from the sample. Then use the standard error to give a \(95 \%\) confidence interval for the proportion of the population to agree with the statement. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. In a random sample of 100 people, 35 agree.

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