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Chapter 8: Problem 9

Sociology: Crime Rate Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980 s and 1990 s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, \(70 \%\) of all arrests are of males aged 15 to 34 years (Source: True Odds by J. Walsh, Merritt Publishing). Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 32 arrests last month, 24 were of males aged 15 to 34 years. Use a \(1 \%\) level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from \(70 \%.\)

Short Answer

Expert verified
No, the test suggests that the proportion is not significantly different from 70%.

Step by step solution

01

Define the Hypotheses

We need to define the null and alternative hypotheses. The null hypothesis, \(H_0\), states that the population proportion \(p\) of arrests of males aged 15 to 34 years is \(0.70\) as it is nationally, so \(H_0: p = 0.70\). The alternative hypothesis, \(H_a\), is that the population proportion is different from \(70\%\), so \(H_a: p eq 0.70\).
02

Determine the Test Statistic

We will use a z-test for proportions since we are dealing with categorical data and a large sample size. The test statistic is given by the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion under the null hypothesis, and \(n\) is the sample size.
03

Calculate the Sample Proportion

The sample proportion \(\hat{p}\) is calculated as the number of arrests of males aged 15 to 34 divided by the total number of arrests: \(\hat{p} = \frac{24}{32} = 0.75\).
04

Calculate the Test Statistic Value

Substitute the values \(\hat{p} = 0.75\), \(p_0 = 0.70\), and \(n = 32\) into the z-formula to find the test statistic: \[ z = \frac{0.75 - 0.70}{\sqrt{\frac{0.70(1-0.70)}{32}}} \]. Calculate this to get \(z \approx 0.87\).
05

Find the Critical Value and Compare

Using a significance level of \(1\%\) for a two-tailed test, the critical z-values are approximately \(-2.576\) and \(2.576\). Since the calculated value \(0.87\) does not exceed the critical values, we do not reject the null hypothesis.
06

Draw a Conclusion

Since the test statistic \(0.87\) falls within the range \(-2.576, 2.576\), we fail to reject the null hypothesis. Thus, there is not enough statistical evidence at the \(1\%\) level of significance to claim that the proportion of arrests of males aged 15 to 34 in Rock Springs differs from \(70\%\).

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Key Concepts

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

Null Hypothesis
The concept of a null hypothesis is central to hypothesis testing. Here, the goal is to have a statement that we try to test, usually reflecting no change or no effect. For the national crime rate, the null hypothesis ( $H_0$) states that the population proportion of arrests of males aged 15 to 34 is exactly equal to the national figure of 70%.
  • Statement Purpose: The null hypothesis assumes no difference. It serves as the starting point for testing, with the idea of disproving it if the data suggests so.
  • Mathematical Representation: In this context, the null hypothesis is expressed as: $H_0: p = 0.70$.
This hypothesis provides a baseline; we presume it to be true until statistical testing suggests otherwise.
Alternative Hypothesis
Opposite to the null hypothesis is the alternative hypothesis (\(a\)), which represents what we aim to support through evidence. In the context of the crime rate, the sociologist is testing if the proportion in Rock Springs is different from the national rate of 70%.
  • Purpose: The alternative hypothesis suggests there is an effect or a difference. It is necessary for determining if the null hypothesis should be rejected.
  • Formulation: For a test where we are interested in any difference, the alternative hypothesis can be two-tailed, expressed as: \(H_a: p eq 0.70\).
Choosing the appropriate alternative hypothesis depends on the research question and hypothesis testing objectives.
Z-Test for Proportions
The Z-test for proportions helps to determine the significance of observing a sample proportion. It is used here because the population is assumed to be normally distributed and we have a large enough sample size.
  • Formula: The Z-test statistic is given by:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized population proportion, and n is the sample size.
  • Interpretation: A Z-test result provides the number of standard deviations the sample proportion is from the hypothesized population proportion. This allows assessing if such an observation is significant or likely due to random chance.
This statistical approach is ideal for evaluating if the local arrest proportion significantly differs from the national crime rate.
Critical Values
Critical values determine the threshold for decision-making in hypothesis testing. They represent the cutoff points of the probability distribution beyond which we reject the null hypothesis.
  • Two-Tailed Test: As the test is two-tailed, the critical values for this test at the 1% level of significance are approximately -2.576 and 2.576.
  • Decision Rule: If the calculated test statistic falls beyond these values in either tail, the null hypothesis is rejected. For example, if $z = 0.87$, it does not exceed the critical bounds, hence the null hypothesis is not rejected.
Critical values help visualize confidence in the sample outcome and guide the final decision on whether to reject or not reject $H_0$.
Significance Level
The significance level in hypothesis testing measures how extreme the data must be to reject the null hypothesis. It controls the probability of committing a Type I error, which means rejecting a true null hypothesis.
  • Chosen Level: A 1% significance level is applied here, indicating a stringent criterion for evidence before rejecting $H_0$. There's only a 1% risk of incorrectly concluding there's a difference when there isn't.
  • Implication: Utilizing a low significance level, such as 1%, means that we require stronger evidence to claim that the local proportion of arrests departs from the national standard.
Choosing an appropriate significance level is crucial for balancing the risks of false positives against the need for sensitivity in the test.

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Most popular questions from this chapter

Thinking Alisha is conducting a paired differences test for a "before \(-(B \text { score })\) and after \((A \text { score })^{\prime \prime}\) situation. She is interested in testing whether the average of the "before" scores is higher than that of the "after" scores. (a) To use a right-tailed test, how should Alisha construct the differences between the "before" and "after" scores? (b) To use a left-tailed test, how should she construct the differences between the "before" and "after" scores?

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Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding \(z\) or \(t\) value as appropriate. (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha ?\) (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom \(d . f .\) not in the Student's \(t\) table, use the closest \(d . f .\) that is smaller. In some situations, this choice of \(d . f .\) may increase the \(P\) -value a small amount and therefore produce a slightly more "conservative" answer. Art Funding: Politics Would you favor spending more federal tax money on the arts? This question was asked by a research group on behalf of The National Institute (Reference: Painting by Numbers, J. Wypijewski, University of California Press). Of a random sample of \(n_{1}=93\) politically conservative voters, \(r_{1}=21\) responded yes. Another random sample of \(n_{2}=83\) politically moderate voters showed that \(r_{2}=22\) responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use \(\alpha=0.05\)

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