91Ó°ÊÓ

A Simple Random Sample, or SRS, is a foundational concept in statistics. It involves selecting a sample in such a way that every individual of the population has an equal chance of being chosen. This method aims to ensure that the sample is representative of the broader population.

The Harris Poll attempted to contact an SRS of smokers by randomly dialing residential telephone numbers. The idea here was that by reaching out to random numbers, each potential respondent—a smoker in this context—would have an equal likelihood of inclusion in the sample.

This method is often used because it helps eliminate selection bias, which might occur if the selection process inadvertently favors certain groups over others. However, it's critical to note that in real-world applications, achieving a truly simple random sample can be challenging due to various factors, such as non-response bias, which we will explore next.

Non-response bias is a type of bias that occurs when certain individuals from a sample do not respond or cannot be reached, and these non-respondents differ significantly from those who do respond.

In the context of the Harris Poll, which used residential phone calls to reach smokers, non-response bias could emerge if certain smokers do not have landlines or are unavailable to answer the phone. Perhaps these individuals have characteristics or habits that differ from those who can be reached, such as a different awareness or concern about the health effects of smoking.

When non-response bias is present, it can skew the results. The opinions of non-respondents might vary significantly from those who participated, potentially leading to inaccurate conclusions about the entire population. As such, acknowledging and attempting to mitigate non-response bias is crucial when designing and interpreting surveys.

The population proportion refers to a percentage or fraction of individuals in a population that share a certain characteristic. In this exercise, the characteristic under study is whether smokers believe that smoking will likely shorten their lives.

From the original sample data, 848 out of 1010 smokers answered "yes" to this question. To find the sample proportion, we use the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successful outcomes and \( n \) is the total number of observations. Substituting the given values, we find \( \hat{p} = \frac{848}{1010} \approx 0.8396 \).

This sample proportion serves as an estimate of the true population proportion. The accuracy of this estimate can be further refined by calculating a confidence interval, which will be discussed in conjunction with the margin of error.

The Margin of Error (ME) is a statistic expressing the amount of random sampling error in a survey's results. It indicates the range within which the true population parameter is expected to lie with a certain level of confidence.

For the Harris Poll, we calculate the margin of error to determine the reliability of our estimate of the population proportion. The formula for ME in a confidence interval for a proportion is determined using the critical value from the standard normal distribution, which is approximately 1.96 for a 95% confidence interval.

To calculate ME, first, we determine the standard error (SE) using \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size. For our data, \( SE \approx 0.0116 \).

• Multiply the SE by the critical value, \( ME = 1.96 \times 0.0116 \approx 0.0228 \).

The resulting ME tells us how much the sample results can be expected to vary from the true population proportion, reinforcing the reliability of the study's conclusions.