Question

Do you think that the United States should pursue a program to send humans to Mars? An opinion poll conducted by the Associated Press indicated that \(49 \%\) of the 1034 adults surveyed think that we should pursue such a program. a. Estimate the true proportion of Americans who think that the United States should pursue a program to send humans to Mars. Calculate the margin of error. b. The question posed in part a was only one of many questions concerning our space program that were asked in the opinion poll. If the Associated Press wanted to report one sampling error that would be valid for the entire poll, what value should they report?

Short answer

Answer: The estimation of the true proportion of Americans who think that the United States should pursue a program to send humans to Mars is 49%, with a margin of error of 3.01%.

Step-by-step solution

a. Finding the true proportion estimation and calculating the margin of error

First, we need to calculate the proportion of positive responses in the poll. \(p = \frac{\text{Number of positive responses}}{\text{Total number of responses}} = \frac{49}{100} = 0.49\) Next, we need to find the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96. Now, let's use the margin of error formula: Margin of Error = \(1.96 * \sqrt{\frac{0.49*(1-0.49)}{1034}} = 1.96 * \sqrt{\frac{0.49*0.51}{1034}} ≈ 0.0301\) The estimation of the true proportion of Americans who think that the United States should pursue a program to send humans to Mars is \(49\%\), with a margin of error of \(3.01\%\).

b. Finding one sampling error for the entire poll

In order to find one sampling error that would be valid for all the questions in the poll, we need to consider the worst-case scenario, which is when the proportion is the most uncertain: when \(p = 0.5\). Now, let's use the margin of error formula with this worst-case proportion value: Margin of Error = \(1.96 * \sqrt{\frac{0.5*(1-0.5)}{1034}} = 1.96 * \sqrt{\frac{0.5*0.5}{1034}} ≈ 0.0304\) The Associated Press should report a single sampling error of \(3.04\%\) as valid for the entire poll.

Key concepts

Proportion Estimation

Proportion estimation is a statistical technique used to approximate the true percentage or ratio within a population based upon sample data. In the context of opinion polls, it involves assessing the portion of respondents who hold a certain view or preference among the total number of individuals polled.

For instance, if an opinion poll reveals that 49% of the surveyed group favors a human mission to Mars, this figure is an estimate of the true proportion of all Americans who share this sentiment. It's important to note that the precision of this estimate improves with a larger sample size and representative sampling methods. Proportion estimation serves as the foundation for calculating the margin of error, which provides a range within which the true proportion is likely to fall.

Confidence Level

The confidence level in statistics denotes the likelihood that the true parameter (such as a proportion or mean) lies within the calculated confidence interval. Commonly expressed as a percentage, it represents how certain we can be about the range in which the true value exists.

For example, a 95% confidence level implies that if the same poll were conducted 100 times under identical conditions, we would expect the true proportion to lie within the estimated interval 95 of those times. The confidence level does not guarantee correctness; rather, it quantifies the level of uncertainty or risk when estimating the population parameter. The choice of confidence level affects the margin of error, where a higher confidence level leads to a wider margin of error, indicating greater uncertainty.

Z-Score

The z-score, also known as a standard score, quantifies the number of standard deviations a data point is from the mean. In the context of confidence intervals, the z-score is used to determine the range within which the true proportion is expected to lie, based on the selected confidence level.

In a 95% confidence interval calculation, the z-score is approximately 1.96, denoting that the true proportion should fall within 1.96 standard deviations from the sample proportion in 95 out of 100 cases. Z-scores vary with confidence levels: higher confidence levels correspond to larger z-scores, and conversely, lower confidence levels correspond to smaller z-scores.

Sampling Error

Sampling error refers to the variation between the estimate derived from a sample and the true value in the population. It arises because the sample is only a subset of the population, thus may not perfectly represent the population's characteristics. Sampling error is an unavoidable aspect of survey research. However, it can be quantified using the margin of error.

For an opinion poll that includes multiple questions, the largest margin of error for any proportion (most often at 50%) is often reported as a conservative estimate of the sampling error. This practice accounts for the highest possible variability to ensure that the concluded margin of error covers all questions within the poll. Dealing with sampling error is crucial for interpreting poll results realistically and understanding the inherent uncertainty in estimations from sample data.