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Chapter 22: Problem 31

Running Red Lights. A random digit dialing telephone survey of 880 drivers asked, "Recalling the last 10 traffic lights you drove through, how many of them were red when you entered the intersections?" Of the 880 respondents, 171 admitted that at least one light had been red. 25 a. Give a \(95 \%\) confidence interval for the proportion of all drivers who ran one or more of the last 10 red lights they met. b. Nonresponse is a practical problem for this survey: only \(21.6 \%\) of calls that reached a live person were completed. Another practical problem is that people may not give truthful answers. What is the likely direction of the bias? Do you think more or fewer than 171 of the 880 respondents really ran a red light? Why?

Short Answer

Expert verified
a. The 95% confidence interval is approximately (0.1678, 0.2208). b. Likely fewer admitted to running red lights than actually did, due to social desirability bias.

Step by step solution

01

Identify the Sample Proportion

First, we need to calculate the sample proportion of drivers who admitted to running at least one red light. There were 171 respondents who admitted this out of 880 total respondents.\[ \hat{p} = \frac{171}{880} \]
02

Calculate the Sample Proportion

Calculate the sample proportion \( \hat{p} \) using the formula from Step 1.\[ \hat{p} = \frac{171}{880} \approx 0.1943 \]
03

Determine the Z-Score for 95% Confidence

For a 95% confidence interval, the Z-score is 1.96. This value corresponds to the critical value for a two-tailed normal distribution where 2.5% is in each tail.
04

Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula:\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]where \( n = 880 \).\[ SE = \sqrt{\frac{0.1943(1 - 0.1943)}{880}} \approx 0.0135 \]
05

Calculate the Confidence Interval

Use the sample proportion and the Z-score to calculate the confidence interval: \[ CI = \hat{p} \pm Z \times SE \]\[ CI = 0.1943 \pm 1.96 \times 0.0135 \approx (0.1678, 0.2208) \]
06

Analyze Nonresponse and Bias

Nonresponse in the survey is high, and only 21.6% of the calls were completed. This suggests that the survey might not be representative of the entire population. Additionally, the social desirability bias is likely to occur, where respondents may underreport running red lights due to the negative perception.
07

Determine Likely Direction of Bias

Considering the biases involved, the likely direction of the bias is that fewer respondents admit to running red lights than actually did. Thus, it is probable that more than 171 out of 880 respondents really ran a red light, but did not report it due to social desirability bias.

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Key Concepts

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

Sample Proportion
The sample proportion is a fundamental concept when analyzing survey data. It represents the fraction of respondents in the sample who exhibit a particular characteristic of interest. In the case of the exercise concerning traffic lights, the sample proportion (\( \hat{p} \)) helps identify the percentage of drivers who admitted to running at least one red light.
  • The formula for calculating the sample proportion is simply the number of respondents admitting to a behavior divided by the total number of respondents surveyed. In this exercise, it's \( \frac{171}{880} \approx 0.1943 \).
  • This indicates that approximately 19.43% of surveyed drivers confessed to running a red light.
The sample proportion is crucial as it forms the basis for further statistical analysis, such as confidence interval estimation, to infer about the larger population of drivers.
Standard Error
The standard error (SE) is essential in determining how much the sample proportion can be expected to vary from the true population proportion. It provides a measure of the precision of the sample estimate.
  • Mathematically, the standard error of the sample proportion is computed using the formula \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] where \( n \) is the sample size.
  • For our exercise, it calculates to approximately \( 0.0135 \), indicating the degree of variation we might expect if the survey were repeated.
A smaller standard error means a more precise estimate, contributing to a more reliable confidence interval for the true population proportion. Understanding standard error helps in assessing the reliability of survey results and anticipating potential deviations.
Bias in Survey
In survey research, biases can significantly impact the results and must be addressed to ensure accurate conclusions. Two common biases relevant to our exercise are:
  • Nonresponse Bias: Since only 21.6% of the calls resulted in completed surveys, those who opted out may differ systematically from those who responded, perhaps influencing the results.
  • Social Desirability Bias: Respondents might under-report behaviors that are socially frowned upon, such as running red lights.
These biases likely result in an underestimation of the actual number of drivers running red lights as fewer might admit this behavior, skewing the survey's findings toward more socially acceptable responses. Identifying and accounting for potential biases is crucial for more accurate representations of the broader population.
Nonresponse
Nonresponse is a critical aspect influencing the validity of survey results. High nonresponse rates can lead to nonresponse bias, where the opinions of those who did not participate may systematically differ from those who did.
  • In this exercise, the nonresponse rate was quite substantial, as only 21.6% of calls were completed.
  • This low completion rate increases the potential for bias because the dataset may not accurately reflect the entire population's behavior or attitudes.
Because of the significant amount of nonrespondents, the conclusions drawn from this survey must be approached with caution, bearing in mind that those who refuse to participate might have different characteristics, such as willingness to admit socially undesirable actions, than those who participated. Understanding the scope of nonresponse helps in evaluating the robustness and generalizability of survey results.

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Most popular questions from this chapter

Gun Violence and Video Games. People disagree about the impact of video games on gun violence. A survey in 2017 of a random sample of 1501 U.S. adults asked "Does the amount of gun violence in video games contribute a great deal or a fair amount to gun violence?" Of the 1501 sampled, 180 said "Not at all."20 a. Give the \(95 \%\) large-sample confidence interval for the proportion \(p\) of all U.S. adults who said that gun violence in video games does not contribute at all to gun violence. Be sure to verify that the sample size is large enough to use the large-sample confidence interval. b. Give the plus four \(95 \%\) confidence interval for \(p\). If you express the two intervals in percentages, rounded to the nearest 10 th of a percent, how do they differ? (The plus four interval always pulls the results toward \(50 \%\).)

Staph Infections. A study investigated ways to prevent staph infections in surgery patients. In a first step, the researchers examined the nasal secretions of a random sample of 6771 patients admitted to various hospitals for surgery. They found that 1251 of these patients tested positive for Staphylococcus aureus, a bacterium responsible for most staph infections. \(.^{4}\) a. Describe the population and explain in words what the parameter \(p\) is. b. Give the numerical value of the statistic \(\widehat{p}\) that estimates \(p\).

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. Only half of all mice complete one particular maze in less than 18 seconds. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures the proportion of 40 mice that completed the maze in less than 18 seconds with noise as a stimulus. The proportion of mice that completed the maze in less than 18 seconds is \(\widehat{p}=0.7\). The hypotheses for a test to answer the researcher's question are a. \(H_{0}: p=0.5, H_{a}: p>0.5\). b. \(H_{0}: p=0.5, H_{a}: p<0.5\). c. \(H_{0}: p=0.5, H_{a}: p \neq 0.5\).

Black Raspberries and Cancer. Sample surveys usually contact large samples, so we can use the large-sample confidence interval if the sample design is close to an SRS. Scientific studies often use smaller samples that require the plus four method. For example, Familial Adenomatous Polyposis (FAP) is a rare inherited disease characterized by the development of an extreme number of polyps early in life and colon cancer in virtually \(100 \%\) of patients before the age of 40 . A group of 14 people suffering from FAP being treated at the Cleveland Clinic drank black raspberry powder in a slurry of water every day for nine months. The numbers of polyps were reduced in 11 out of 14 of these patients. 19 a. Why can't we use the large-sample confidence interval for the proportion \(p\) of patients suffering from FAP that will have the number of polyps reduced after nine months of treatment? b. The plus four method adds four observations: two successes and two failures. What are the sample size and the number of successes after you do this? What is the plus four estimate \(\tilde{p}\) of \(p\) ? c. Give the plus four \(90 \%\) confidence interval for the proportion of patients suffering from FAP who will have the number of polyps reduced after nine months of treatment.

Coin Tossing. The French naturalist Count Buffon (1707-1788) tossed a coin 4040 times. The result was 2048 heads, with \(\frac{2048}{4040}=0.5069\) the proportion of heads. Is this evidence that the coin was not fair? State the appropriate hypotheses and give the P-value.
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