Question

The article "Viewers Speak Out Against Reality TV" (Associated Press, September 12,2005 ) included the following statement: "Few people believe there's much reality in reality TV: a total of 82 percent said the shows are either 'totally made up' or 'mostly distorted'." This statement was based on a survey of 1002 randomly selected adults. Compute and interpret a bound on the error of estimation for the reported percentage.

Short answer

To calculate the bound of the error of estimation for the reported percentage in a survey, use the formula for standard error of a proportion, substituting the given proportion and sample size. The resultant standard error represents the bound on the error of estimation. Calculate this to obtain a numerical value.

Step-by-step solution

Recall the formula for standard error of a proportion

The standard error of a proportion is defined by the formula \(SE = \sqrt{p(1-p)/n}\), where \(p\) is the proportion (in this case, 0.82) and \(n\) is the sample size (in this case, 1002).

Insert the values into the formula and calculate

Insert the given values into the standard error formula: \(SE = \sqrt{0.82(1-0.82)/1002} \). Carry out the calculation to find the standard error which can give a rough approximation of the bound on the error.

Interpretation

The standard error calculated in the previous step is a measure of the accuracy of our sample proportion in estimating the true population proportion. The smaller the standard error, the more accurate the estimate. It represents the standard deviation of the sampling distribution of the sample proportion. This calculated standard error then serves as an estimate of the bound for the error of estimation for the reported percentage.

Key concepts

Standard Error

When we talk about standard error, we refer to a statistic that quantifies the precision with which a sample estimate approximates the true population parameter. It's important because it gives us insight into the reliability of the estimate obtained from our sample data. In the context of proportions, the formula becomes \[\begin{equation}SE = \sqrt{\frac{p(1-p)}{n}}\end{equation}\]where p is the sample proportion and n is the sample size. If you were dealing with the average of some measurements, you would use a different version of the formula, one involving the standard deviation and the square root of the sample size.

• The smaller the standard error, the closer your sample proportion is likely to be to the true population proportion.
• A larger sample size will generally lead to a smaller standard error, making your estimate more reliable.
• The standard error can be influenced by the variability of the population—the more varied the population, the larger the standard error might be, even with a large sample size.

Understanding the standard error is crucial for interpreting survey results and can help you judge the margin of error for the estimates provided.

Sample Proportion

The sample proportion is a statistic that represents the fraction of the sample with a certain characteristic. In our exercise, it relates to the proportion of adults who think that reality TV shows are either 'totally made up' or 'mostly distorted'. The given proportion (p) is 0.82 or 82%, meaning that out of the randomly selected adults, 82% shared this opinion about reality TV.

Why is this important?

• Sample proportions are used as estimates for the true population proportion.
• The accuracy of the sample proportion as an estimator depends on the sample size and the variability within the population.
• With a larger sample size, the sample proportion typically becomes a better estimator of the population proportion.

This specific measure is vital to obtaining an accurate picture of public opinion. It's what pollsters use to gauge the sentiments of the broader population based on a smaller group of individuals. That said, it's not without its limitations; sampling error and the potential bias in how the sample was collected could affect its accuracy.

Sampling Distribution

Lastly, the sampling distribution is a theoretical distribution that represents how the sample statistic would behave if we were to take many samples from the same population.

Key aspects:

• It's the probability distribution of a given statistic based on a random sample.
• The sampling distribution of the sample proportion can tell us how the proportion will vary from sample to sample.
• The shape of this distribution is often approximately normal when the sample size is large, thanks to the Central Limit Theorem.

For the exercise in question, this concept allows us to understand the variability and therefore the reliability of the sample proportion of 82%. The standard error of the sample proportion, which we calculated, can be viewed as the standard deviation of its sampling distribution. If we assume a normal distribution, we can use this to create confidence intervals around the sample proportion to make educated guesses about the true population proportion. This ties back into the concept of the error of estimation, giving us insight into how much the sample proportion may differ from the actual population proportion.