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Chapter 10: Problem 4

Do state laws that allow private citizens to carry concealed weapons result in a reduced crime rate? The author of a study carried out by the Brookings Institution is reported as saying, 'The strongest thing I could say is that \(\bar{I}\) don't see any strong evidence that they are reducing crime" (San Luis Obispo Tribune, January 23, 2003). a. Is this conclusion consistent with testing \(H_{0}:\) concealed weapons laws reduce crime versus \(H_{a}\) : concealed weapons laws do not reduce crime or with testing \(H_{0}\) : concealed weapons laws do not reduce crime versus \(H_{e}:\) concealed weapons laws reduce crime Explain. b. Does the stated conclusion indicate that the null hypothesis was rejected or not rejected? Explain.

Short Answer

Expert verified
a. The correct pair of hypotheses for this problem is \(H_{0}\): 'concealed weapons laws do not reduce crime' and \(H_{a}\): 'concealed weapons laws reduce crime'. b. Based on the researcher's comment, the null hypothesis was not rejected because he did not find sufficient evidence against it.

Step by step solution

01

Identifying the Hypotheses

Firstly, it is critical to identify the correct pair of hypotheses. Since null hypothesis (\(H_{0}\)) is a statement of no effect or no difference which we assume to be true until we have enough evidence to the contrary, in this context it should be 'concealed weapons laws do not reduce crime'. The alternative hypothesis (\(H_{a}\)) is then what we might believe if we find ample evidence against \(H_{0}\), which in this case would be 'concealed weapons laws reduce crime'.
02

Interpreting the Result

The second part is to determine whether the null hypothesis was rejected or not based on the researcher's comment. The researcher stated, 'The strongest thing I could say is that I don't see any strong evidence that they are reducing crime'. This implies that the researcher did not find sufficient evidence to reject the null hypothesis. Therefore, the null hypothesis has not been rejected.

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Key Concepts

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

Null Hypothesis
The null hypothesis is an essential part of hypothesis testing. It represents a statement of no effect or no difference, and it serves as the starting point for any statistical test.
For our exercise, the null hypothesis (\(H_0\)) is stated as "concealed weapons laws do not reduce crime."
When conducting a hypothesis test, we assume the null hypothesis is true until we have enough statistical evidence to prove otherwise.

Some key points to consider about the null hypothesis include:
  • It provides a benchmark for comparison.
  • It allows us to determine if observed data can be explained by chance.
  • Refuting it requires strong evidence.
In the context of the exercise, if data indicates that the crime rate remains unchanged with concealed weapons laws, this supports the null hypothesis.
But if evidence shows a significant reduction, then the null hypothesis might be rejected. In the reported study, the researcher did not find such convincing evidence, suggesting the null hypothesis was not rejected.
Alternative Hypothesis
The alternative hypothesis (\(H_a\)) stands in contrast to the null hypothesis. It is what researchers aim to support through their analysis. It indicates the presence of an effect or difference that experiments seek to validate.
In this scenario, it would be stated as "concealed weapons laws reduce crime."
This hypothesis proposes that there is a real effect of concealed weapons laws on crime reduction.

Here's what to remember about the alternative hypothesis:
  • It suggests an outcome that goes against the null hypothesis.
  • It becomes the focus if the null hypothesis is refuted.
  • Evidence must be strong and convincing to support this hypothesis.
If significant evidence suggests crime reduction linked to these laws, we may reject the null hypothesis in favor of this alternative.
However, given the researcher's comment, there wasn't strong evidence to affirm this alternative hypothesis in their analysis.
Statistical Evidence
Statistical evidence is the crux of hypothesis testing, providing the necessary data to decide between the null and alternative hypotheses.
It involves analyzing sample data and determining whether observed effects are strong enough to challenge the null hypothesis.

The role of statistical evidence includes:
  • Quantifying the probability of observed data under the null hypothesis.
  • Guiding decisions about the rejection or acceptance of hypotheses.
  • Using tools like p-values and confidence intervals to gauge the strength of evidence.
The comment from the researcher鈥�"I don't see any strong evidence that they are reducing crime"鈥攕uggests that their statistical analysis didn't offer compelling proof to dispute the null hypothesis.
Hence, the statistical evidence was likely not strong enough to reject the notion that "concealed weapons laws do not reduce crime."
This emphasizes the importance of robust statistical evidence in reaching conclusions in scientific studies.

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Most popular questions from this chapter

The success of the U.S. census depends on people filling out and returning census forms. Despite extensive advertising, many Americans are skeptical about claims that the Census Bureau will guard the information it collects from other government agencies. In a USA Today poll (March 13,2000 ), only 432 of 1004 adults surveyed said that they believe the Census Bureau when it says the information you give about yourself is kept confidential. Is there convincing evidence that, despite the advertising campaign, fewer than half of U.S. adults believe the Census Bureau will keep information confidential? Use a significance level of \(.01\).

Teenagers (age 15 to 20 ) make up \(7 \%\) of the driving population. The article "More States Demand Teens Pass Rigorous Driving Tests" (San Luis Obispo Tribune, January 27,2000 ) described a study of auto accidents conducted by the Insurance Institute for Highway Safety. The Institute found that \(14 \%\) of the accidents studied involved teenage drivers. Suppose that this percentage was based on examining records from 500 randomly selected accidents. Does the study provide convincing evidence that the proportion of accidents involving teenage drivers differs from \(.07\), the proportion of teens in the driving population?

Optical fibers are used in telecommunications to transmit light. Current technology allows production of fibers that transmit light about \(50 \mathrm{~km}\) (Research at Rensselaer, 1984 ). Researchers are trying to develop a new type of glass fiber that will increase this distance. In evaluating a new fiber, it is of interest to test \(H_{0}: \mu=50\) versus \(H_{a}: \mu>50\), with \(\mu\) denoting the true average transmission distance for the new optical fiber. a. Assuming \(\sigma=10\) and \(n=10\), use Appendix Table 5 to find \(\beta\), the probability of a Type II error, for each of the given alternative values of \(\mu\) when a level \(.05\) test is employed: \(\begin{array}{llll}\text { i. } 52 & \text { ii. } 55 & \text { iii. } 60 & \text { iv. } 70\end{array}\) b. What happens to \(\beta\) in each of the cases in Part (a) if \(\sigma\) is actually larger than \(10 ?\) Explain your reasoning.

The state of Georgia's HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, \(53.2 \%\) had a GPA below \(3.0\) at the end of their first year ("Who Loses HOPE? Attrition from Georgia's College Scholarship Program," Southern Economic Journal [1999]: \(379-390\) ). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

A hot tub manufacturer advertises that with its heating equipment, a temperature of \(100^{\circ} \mathrm{F}\) can be achieved in at most 15 min. A random sample of 25 tubs is selected, and the time necessary to achieve a \(100^{\circ} \mathrm{F}\) temperature is determined for each tub. The sample average time and sample standard deviation are \(17.5 \mathrm{~min}\) and \(2.2 \mathrm{~min}, \mathrm{re}-\) spectively. Does this information cast doubt on the company's claim? Carry out a test of hypotheses using significance level \(.05 .\)
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