Question

A recent biological study performed on a large random sample of North American birds found that \(46 \%\) of birds' nests experienced full or partial nest predation, in which some type of predator raided the nest before the hatchlings had flown off. The margin of error for the study was \(3 \%\). Which of the following is the best interpretation of the margin of error for this study? A) The percentage of North American birds whose nests get raided by predators is likely somewhere between \(43 \%\) and \(49 \%\). B) There is a \(3 \%\) chance that the study's finding about nest predation for North American birds is incorrect. C) It is unlikely that fewer than \(43 \%\) of birds' nests will get raided this coming year. D) The research indicates that no more than \(49 \%\) of birds' nests will likely get raided by predators.

Short answer

The best interpretation of the margin of error for this study is that the percentage of North American birds whose nests get raided by predators is likely somewhere between \(43\%\) and \(49\%\).

Step-by-step solution

1. Understand Margin of Error

Margin of error is a range that gives a level of confidence that the true value lies within this range. In this case, it is given as 3%.

2. Calculate the Range

To find the range within which the true percentage of nests experiencing predation lies, we add and subtract the margin of error from the given percentage (46%). Lower bound: \(46\% - 3\% = 43\% \newline \) Upper bound: \(46\% + 3\% = 49\% \newline \) So, the range is 43% to 49%.

3. Analyze the Answer Choices

Now, let's analyze each option to determine the best interpretation of the margin of error: A) This choice correctly presents the range (43% to 49%) within which the true percentage of nests experiencing predation is likely to lie. B) The 3% margin of error does not represent the chance the study's findings are incorrect; it represents the range around the given percentage. C) This choice focuses on the lower bound of the range but does not provide information on the upper bound, which is also crucial for the correct interpretation of the margin of error. D) Like choice C, this option emphasizes the upper bound but does not include the lower bound, making it an incorrect interpretation of the margin of error. The correct answer is A.

Key concepts

Statistical Confidence

Understanding the concept of statistical confidence is essential when interpreting data from studies and surveys. It tells us how certain we can be that the results from a sample accurately reflect the entire population. In the exercise's context, the margin of error is closely related to statistical confidence. It provides a range that expresses the degree of confidence that the true value (the percentage of birds' nests experiencing predation) is within a certain interval around the observed statistic.

In practical terms, if the margin of error is 3%, and our sample statistic is 46%, we can be confident (typically 95% confident, although the specific confidence level is not stated in the problem) that the actual population parameter lies between 43% and 49%. However, it's essential to remember that statistical confidence does not imply certainty; it is a measure of how likely it is that our interval contains the true value, not a guarantee.

Probability and Statistics

Probability and statistics are foundational concepts that students must grasp to perform well in SAT math preparation and other academic pursuits. These two concepts allow us to analyze and make predictions based on data. Probability measures the likelihood of an event occurring, whereas statistics is the science of collecting, analyzing, interpreting, presenting, and organizing data.

In the given exercise, understanding the principles of probability would help students realize that the margin of error does not represent the probability of the study being incorrect. Instead, it's a statistical term that reflects the expected variability when the study is repeated under the same conditions. It's essential to differentiate between the likelihood of an event (probability) and the confidence with which we can expect a range to contain the true value (statistical confidence), as they are related but distinctly different concepts.

Data Interpretation

Data interpretation is a crucial skill for analyzing the results of studies and making informed decisions. It involves extracting meaningful insights from data and understanding how to apply them to real-world situations. The exercise showcases the importance of correctly interpreting the margin of error when looking at study findings. An incorrect interpretation can lead to misguided conclusions about the data.

Many students might find it difficult to distinguish between the various interpretations of the margin of error provided in the exercise options. It's crucial to recognize that the margin of error provides a symmetric range around the observed statistic (in this case, 46%), rather than suggesting an exact limit or a singular probability of an incorrect outcome. Clear and accurate data interpretation skills can help prevent such misunderstandings and enable students to make better use of statistical information.

SAT Math Preparation

Preparing for the SAT math section requires not only proficiency in mathematical computation but also an understanding of how that math applies to different contexts, including data analysis. The margin of error concept discussed in the provided exercise is a typical example of a SAT-style question that tests students' ability to apply statistical knowledge in a practical scenario.

For effective SAT math preparation, students should focus on mastering key topics, such as probability, statistics, margin of error, and statistical confidence. Engaging with exercises that present real-world applications of these concepts will enhance their comprehension and boost their test performance. Critical to this is the interpretation of data, for which students must read questions carefully and understand what is being asked, as exemplified by the need to identify the correct interpretation of the margin of error in different answer choices.