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Statistics is a branch of mathematics that involves the collection, analysis, interpretation, presentation, and organization of data. It serves as a fundamental tool in various fields to make informed decisions based on empirical data.
When you hear about a percentage, like 68% of U.S. adults having an opinion according to a survey, it's statistics in action. This percentage comes from collecting data points through surveys or experiments.

• Data Collection: Information is gathered through various means like surveys, experiments, and observations.
• Data Interpretation: This involves understanding what the data tells us and making sense of the figures we see.
• Probability: In statistics, probability provides a measure of how likely something is to occur. For instance, determining the likelihood of respondents sharing an opinion based on survey results.

In the context of the given exercise, statistics were utilized to understand the opinions of U.S. adults concerning government action on water quality. This understanding is instrumental in making inferences about the general population based on sample data.

Surveys are one of the most common tools in statistics to collect data from a predefined group, with the aim to understand the opinions, behaviors, or characteristics of a larger population.
Surveys collect responses that are then converted into statistical figures to allow for data analysis.

• Design: Surveys need to be carefully crafted to collect relevant and unbiased data.
• Sampling: Survey respondents should be representative of the wider population to ensure accurate results.
• Margin of Error: This symbolizes the extent of random sampling error in the survey results, which, in the exercise, is given as ±1.6%.

In the exercise scenario, the survey conducted by Pew Research represents the opinions of U.S. adults about government policies. The margin of error tells us how much these results might differ from the true population opinions.

The complement rule is a fundamental concept in probability theory that deals with the likelihood of an event not occurring. The probability of an event happening plus the probability of it not happening always equals 1.
This principle helps in calculating probabilities in scenarios where understanding the non-occurrence is easier or more intuitive.

• Complement: The complement of an event A is defined as all outcomes in the sample space that are not part of A.
• Formula: If the probability of an event occurring is p, then the probability of the event not occurring is (1 - p).
• Application: It is particularly useful in complex probability calculations where direct computation might be challenging.

In the exercise, finding the probability that neither selected adults agrees required using the complement rule. Since 68% believe the government is doing too little, the complement probability, 32%, represents those who don't.