Question

A reporter on cnn.com stated in July 2010 that \(95 \%\) of all court cases that go to trial result in a guilty verdict. To test the accuracy of this claim, we collect a random sample of 2000 court cases that went to trial and record the proportion that resulted in a guilty verdict. (a) What is/are the relevant parameter(s)? What sample statistic(s) is/are used to conduct the test? (b) State the null and alternative hypotheses. (c) We assess evidence by considering how likely our sample results are when \(H_{0}\) is true. What does that mean in this case?

Short answer

(a) The relevant parameter is the population proportion \(p\) of guilty verdicts in all court cases, with sample proportion \(\hat{p}\) for the assessment. (b) The null hypothesis is \(H_{0}\): \(p = 0.95\) and the alternative hypothesis is \(H_{a}\): \(p \neq 0.95\). (c) The assessment involves determining how likely the observed sample result would be if \(H_{0}\) (95% of court cases result in a guilty verdict) is true.

Step-by-step solution

Identifying relevant parameters and statistics

The relevant parameter in this scenario is the population proportion of all court cases that result in a guilty verdict. Let's denote this as \(p\). The sample statistic used to conduct the test is the sample proportion of court cases sampled that resulted in a guilty verdict, denoted as \(\hat{p}\).

Formulating the null and alternative hypotheses

The null hypothesis, \(H_{0}\), is a statement that the parameter, \(p\), is equal to a specific value. The alternative hypothesis, \(H_{a}\), states that \(p\) takes on a value that is different, larger, or smaller than the value specified in \(H_{0}\). Here, the null hypothesis \(H_{0}\) : \(p = 0.95\) (95% of all cases are declared guilty, according to the reporter). The alternate hypothesis could assume that the proportion is different from 0.95, so \(H_{a}\): \(p \neq 0.95\).

Understanding the Assessment

The sentence 'We assess evidence by considering how likely our sample results are when \(H_{0}\) is true' means that we would think about how probable it would be to obtain our sample data if indeed 95% of all court cases resulted in a guilty verdict. If the sample data is very unusual assuming the null hypothesis is true, we would reject \(H_{0}\) in favor of \(H_{a}\).

Key concepts

Population Proportion

In the context of hypothesis testing, the population proportion refers to a specific characteristic of an entire population, such as the percentage of court cases that result in a guilty verdict. It's denoted as \( p \) and is a fixed value, although unknown. When we seek to estimate the population proportion, we are trying to find what percentage of all possible cases, not just our sample, would end in a similar outcome. If the population is the pool of all court cases that go to trial, then the population proportion is the overall percentage of these cases that lead to a guilty verdict.

Sample Statistic

A sample statistic is a numerical summary about a sample, which serves as an estimate for the unknown population parameter. In our court case scenario, the relevant sample statistic is the sample proportion, \( \hat{p} \) , which is the percentage of cases in our selected sample that resulted in a guilty verdict. By comparing this \( \hat{p} \) with the claimed population proportion, we can conduct a hypothesis test to assess the claim's accuracy. Sample statistics vary from sample to sample due to variability in the selection process, which is a crucial concept in statistics known as sampling variability.

Null Hypothesis

The null hypothesis, symbolized as \( H_{0} \) , is a statement for a statistical hypothesis test that assumes no effect or no difference. It represents the skeptical perspective, providing a benchmark against which we measure the evidence provided by our sample statistic. In our case, the null hypothesis states that the true population proportion of guilty verdicts in court cases (\( p \) ) is equal to 95%, \( H_{0} : p = 0.95 \) . If the data collected from our sample significantly deviates from this assumption, we might reject the null hypothesis in favor of the alternative.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_{a} \) or \( H_{1} \) , is a statement that proposes a new effect, relationship, or difference, contrasting the skeptical stance of the null hypothesis. For our exercise, the alternative hypothesis presents the possibility that the population proportion (\( p \) ) is not 95%. Formally, we write \( H_{a}: p eq 0.95 \) . It reflects the belief that what was stated by the reporter might be incorrect and allows us to challenge the status quo presented by \( H_{0} \) with the sample evidence.

Evidence Assessment in Statistics

In evidence assessment, we use our collected sample data to evaluate how consistent it is with the null hypothesis. This process involves calculating the probability of observing our sample statistic, like the sample proportion of guilty verdicts, given that the null hypothesis is true. If this probability, known as a p-value, is very low, it suggests that our sample result is unlikely under the null hypothesis, and thus, we may have enough evidence to reject \( H_{0} \) in favor of \( H_{a} \) . In our example, if the likelihood of obtaining our sample's guilty verdict proportion is very small under the assumption that the actual population proportion is 95%, this would cast doubt on the accuracy of the reporter's claim.