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Chapter 4: Problem 18

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

Expert verified
(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

01

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 \(\hat{p}\).
02

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 \(H_{a}\): \(p \neq 0.95\).
03

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 \(H_{a}\).

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Key Concepts

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

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.

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Most popular questions from this chapter

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2} .\) In addition, in each case for which the results are significant, state which group ( 1 or 2 ) has the larger mean. (a) \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : 0.12 to 0.54 (b) \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -2.1 to 5.4 (c) \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) : -10.8 to -3.7

Test \(\mathrm{A}\) is described in a journal article as being significant with " \(P<.01\) "; Test \(\mathrm{B}\) in the same article is described as being significant with " \(P<\).10." Using only this information, which test would you suspect provides stronger evidence for its alternative hypothesis?

It is well established that exercise is beneficial for our bodies. Recent studies appear to indicate that exercise can also do wonders for our brains, or, at least, the brains of mice. In a randomized experiment, one group of mice was given access to a running wheel while a second group of mice was kept sedentary. According to an article describing the study, "The brains of mice and rats that were allowed to run on wheels pulsed with vigorous, newly born neurons, and those animals then breezed through mazes and other tests of rodent IQ"9 compared to the sedentary mice. Studies are examining the reasons for these beneficial effects of exercise on rodent (and perhaps human) intelligence. High levels of BMP (bonemorphogenetic protein) in the brain seem to make stem cells less active, which makes the brain slower and less nimble. Exercise seems to reduce the level of BMP in the brain. Additionally, exercise increases a brain protein called noggin, which improves the brain's ability. Indeed, large doses of noggin turned mice into "little mouse geniuses," according to Dr. Kessler, one of the lead authors of the study. While research is ongoing in determining how strong the effects are, all evidence points to the fact that exercise is good for the brain. Several tests involving these studies are described. In each case, define the relevant parameters and state the null and alternative hypotheses. (a) Testing to see if there is evidence that mice allowed to exercise have lower levels of BMP in the brain on average than sedentary mice. (b) Testing to see if there is evidence that mice allowed to exercise have higher levels of noggin in the brain on average than sedentary mice. (c) Testing to see if there is evidence of a negative correlation between the level of BMP and the level of noggin in the brains of mice.

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing a new drug with potentially dangerous side effects to see if it is significantly better than the drug currently in use. If it is found to be more effective, it will be prescribed to millions of people.

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if there is evidence that the proportion of people who smoke is greater for males than for females.
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