91Ó°ÊÓ

A survey of licensed drivers inquired about running red lights. One question asked, "Of every 10 motorists who run a red light, about how many do you think will be caught?" The mean result for 880 respondents was \(\mathrm{x}^{-} \bar{x}\) \(=1.92\) and the standard deviation was \(s=1.83 .{ }^{2}\) For this large sample, \(s\) will be close to the population standard deviation \(\sigma\), so suppose we know that \(\sigma=1.83\). (a) Give a 95\% confidence interval for the mean opinion in the population of all licensed drivers. (b) The distribution of responses is skewed to the right rather than Normal. This will not strongly affect the \(z\) confidence interval for this sample. Why not? (c) The 880 respondents are an SRS from completed calls among 45,956 calls to randomly chosen residential telephone numbers listed in telephone directories. Only 5029 of the calls were completed. This information gives two reasons to suspect that the sample may not represent all licensed drivers. What are these reasons?

Step by step solution

01

Identify Known Values

We are given the following information: the sample mean \( \bar{x} = 1.92 \), the sample size \( n = 880 \), and the population standard deviation \( \sigma = 1.83 \).

02

Find the Critical Z-value for 95% Confidence Interval

For a 95% confidence interval, the critical z-value (\( z^* \)) corresponding to a two-tailed test is 1.96, which can be found using statistical tables or a calculator.

03

Calculate Standard Error of the Mean (SEM)

The Standard Error of the Mean is calculated using the formula: \( SEM = \frac{\sigma}{\sqrt{n}} = \frac{1.83}{\sqrt{880}} \approx 0.0618 \).

04

Calculate Margin of Error

The Margin of Error (ME) is given by the formula: \( ME = z^* \times SEM = 1.96 \times 0.0618 \approx 0.1211 \).

05

Construct Confidence Interval

The 95% confidence interval is expressed as: \( \bar{x} \pm ME = 1.92 \pm 0.1211 \). Therefore, the interval is approximately (1.7989, 2.0411).

06

Explain Influence of Sample Size on Confidence Interval

For question (b), the confidence interval is not strongly affected because a large sample size (n = 880) tends to produce a sampling distribution of the sample mean that is approximately normal, even if the population distribution is skewed, due to the Central Limit Theorem.

07

Discuss Sampling Bias

For question (c), two reasons the sample may not represent all licensed drivers are: (1) the use of landline telephone directories excludes unlisted and mobile-only households, and (2) the response rate is low, as only 5029 calls were completed out of 45956 attempts. This could result in non-response bias.

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.

Central Limit Theorem

The Central Limit Theorem (CLT) is a cornerstone concept in statistics. It explains that when we take large random samples from a population, the distribution of the sample means will tend to be normal, even if the original population distribution is not. This is critical because it allows us to make inferences about the population mean from the sample mean.
For example, even if the responses about running red lights are skewed, with a large enough sample size, the mean of the survey responses will approximate a normal distribution.

• The sample size in our survey is 880, which is large enough for the CLT to hold true.
• Through the CLT, we are able to construct confidence intervals and conduct hypothesis tests effectively.

This is why the skewness of the distribution does not drastically affect our confidence interval calculation.

Standard Error

The Standard Error (SE) of a statistic is a measure of the statistical accuracy of an estimate. It's essentially the standard deviation of its sampling distribution.
For the mean, the standard error of the mean is calculated as \(SEM = \frac{\sigma}{\sqrt{n}}\). In our survey, this translates to \(SEM = \frac{1.83}{\sqrt{880}} \approx 0.0618\).

• SE indicates how much the sample mean would vary if you were to repeat a survey numerous times.
• It helps in gauging the reliability of the sample mean as an estimate of the population mean.

A smaller SE suggests that the sample mean is a more precise estimate of the population mean.

Sampling Bias

Sampling bias occurs when some members of the intended population are less likely to be included in the sample than others.
In this survey, there are two primary sources of potential sampling bias:

• Exclusive reliance on landline telephone directories excludes households that are unlisted or exclusively use mobile phones. This might miss a significant portion of the population who may have different opinions on the topic.
• A low response rate: Out of 45,956 calls, only 5029 were completed, resulting in a low response rate. This might lead to non-response bias, as those who chose not to respond might systematically differ from those who did.

Addressing these biases is essential for ensuring that the survey results are reliable and representative of the entire population of licensed drivers.

Survey Methodology

Survey methodology involves the processes and techniques used to collect data from respondents. It plays a crucial role in the validity and reliability of the survey results.
In the case of our survey on running red lights, the methodology could be improved in several ways:

• Enhancing the sampling frame to include mobile phone users and other traditionally underrepresented groups would help in gathering a more representative sample.
• Improving the response rate, perhaps by employing follow-up calls or offering incentives, could reduce non-response bias.

By refining the survey methodology, the estimates of the population's opinions about running red lights can become more accurate, leading to better-informed decisions and policies.