91Ó°ÊÓ

The article "CSI Effect Has Juries Wanting More Evidence" (USA Today, August 5,2004 ) examines how the popularity of crime-scene investigation television shows is influencing jurors' expectations of what evidence should be produced at a trial. In a survey of 500 potential jurors, one study found that 350 were regular watchers of at least one crime-scene forensics television series. a. Assuming that it is reasonable to regard this sample of 500 potential jurors as representative of potential jurors in the United States, use the given information to construct and interpret a \(95 \%\) confidence interval for the true proportion of potential jurors who regularly watch at least one crime-scene investigation series. b. Would a 99\% confidence interval be wider or narrower than the \(95 \%\) confidence interval from Part (a)?

Step by step solution

01

Calculate the sample proportion

The sample proportion (p) is calculated by dividing the number of successful outcomes (350 jurors watching the crime-scene investigation series) by the total number of outcomes (500 jurors). So, \(p = 350/500 = 0.7\)

02

Apply the formula to calculate the 95% confidence interval

We apply the formula \(p \pm z \sqrt {p(1-p)/n}\) for the 95% confidence interval. Substituting the values, we get \(0.7 \pm 1.96 \sqrt {0.7 \times 0.3/500}\). After calculation, we get approximately \(0.7 \pm 0.042\).

03

Interpret the 95% confidence interval

So, we are 95% confident that the true proportion of potential jurors who regularly watch at least one crime-scene investigation series lies between 65.8% (0.7 - 0.042) and 74.2% (0.7 + 0.042).

04

Compare the 95% confidence interval with a 99% confidence interval

In general, a higher degree of confidence demands a higher level of assurance that the parameter lies within the specified range. Hence, it would require us to expand our range or in other words, a 99% confidence interval would be wider than a 95% confidence interval.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Proportion

The sample proportion is the part of a total population that shares a certain attribute or characteristic. In this exercise, it helps us understand how many potential jurors regularly watch crime-scene investigation shows. By dividing the number of jurors who watch these shows (350) by the total number of jurors surveyed (500), we calculate this sample proportion.

• Formula for sample proportion: \[ p = \frac{x}{n} \]where \(x\) is the number of successful outcomes and \(n\) is the total number of trials.
• For the given data: \(p = \frac{350}{500} = 0.7\).

This means that 70% of surveyed jurors are fans of crime-scene shows. This figure helps us estimate the overall interest among potential jurors in the general population.

Z-Score

A z-score is a statistical measure that tells us how many standard deviations an element is from the mean. It's crucial in constructing confidence intervals. In this context, we use a z-score of 1.96 for the 95% confidence interval, as it represents the normal distribution's critical values.

• Z-score values differ with confidence levels: - For 95% confidence, it's about 1.96.- For 99% confidence, it increases to approximately 2.576.
• Role in the confidence interval formula: The z-score helps determine the margin of error:\[ \text{Margin of Error} = z \cdot \sqrt{\frac{p(1-p)}{n}} \]
• A higher z-score widens the confidence interval because it provides a greater assurance that the true parameter lies within it.

Using the right z-score ensures that our predictions are both precise and reliable.

Statistical Inference

Statistical inference allows us to make conclusions about a larger population based on sample data. It's vital in research and decision-making processes. In this exercise, it enables us to deduce the viewing habits of all potential jurors from the sampled group.

• Key components of statistical inference:- **Point Estimate:** Our sample proportion of 0.7 serves as an estimate of the true population proportion.- **Confidence Interval:** Provides a range in which the true proportion is likely to fall. This range reflects uncertainty in our point estimate.
• The confidence interval is computed using the formula:\[ p \pm z \cdot \sqrt{\frac{p(1-p)}{n}} \]
• Benefits: Such inferences save time and resources since examining the entire population is often impractical.They also provide a scientific basis for predictions.

By using statistical inference, we become more informed about trends and behaviors on a broad scale.

Crime-Scene Investigation Influence on Jurors

The influence of crime-scene investigation shows on jurors is a fascinating example of how media can shape perceptions and expectations. With crime scene shows often glamourizing the intricacies of forensic science, jurors might expect rigorous forensic evidence in real-life trials.

• **Media Influence:** Such shows emphasize detailed, scientific methods for solving crimes, leading jurors to expect similar standards in court.
• **Survey Impact:** The survey from the exercise, where 70% watch such shows, underlines the widespread nature of this influence.
• **Jury Expectations:** As crime-scene shows become cultural staples, jurors might demand more thorough evidence. This can affect trial proceedings and outcomes.

Understanding such influences allows legal professionals to address and manage jurors' expectations better, ensuring fair trial outcomes.