Question

According to the EPA, chloroform, which in its gaseous form is suspected of being a cancer-causing agent, is present in small quantities in all of the country's 240,000 public water sources. If the mean and standard deviation of the amounts of chloroform present in the water sources are 34 and 53 micrograms per liter, respectively, describe the distribution for the population of all public water sources.

Short answer

Answer: We can describe the distribution as a normal distribution with a mean (μ) of 34 micrograms per liter and a standard deviation (σ) of 53 micrograms per liter. Approximately 68% of the water sources have amounts of chloroform within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Step-by-step solution

Understand the problem

We have a population of 240,000 public water sources, and we know the mean and standard deviation of the amounts of chloroform in these water sources. We want to describe the distribution of the amounts of chloroform present in the entire population of public water sources.

Identify the given information

The mean (μ) and standard deviation (σ) of the amounts of chloroform present in the water sources are given as follows: μ = 34 micrograms per liter σ = 53 micrograms per liter

Describe the distribution

Since we have the mean and standard deviation for the population, we can use this information to describe the distribution. We can assume that the distribution of amounts of chloroform in public water sources follows a normal distribution. A normal distribution is characterized by its mean (μ) and standard deviation (σ), and it has a bell-shaped curve that is symmetric about the mean.

State the characteristics of the distribution

Since the distribution is assumed to be normal, we can describe it using the mean and standard deviation values. The distribution of the amounts of chloroform present in the population of 240,000 public water sources can be described as follows: - It follows a normal distribution with a mean (μ) of 34 micrograms per liter and a standard deviation (σ) of 53 micrograms per liter. - Approximately 68% of the water sources have amounts of chloroform within one standard deviation from the mean (i.e., between [34 - 53] and [34 + 53] micrograms per liter). - Approximately 95% of the water sources have amounts of chloroform within two standard deviations from the mean (i.e., between [34 - 2*53] and [34 + 2*53] micrograms per liter). - Approximately 99.7% of the water sources have amounts of chloroform within three standard deviations from the mean (i.e., between [34 - 3*53] and [34 + 3*53] micrograms per liter).

Key concepts

Mean

The "mean" is a measure that gives us an average value of a data set. When we talk about the mean in the context of a normal distribution, it's denoted by the Greek letter \( \mu \). This value is what we expect from a typical observation in our dataset. Here, the mean is 34 micrograms per liter. That means, on average, the amount of chloroform present in the population's water sources is 34 micrograms per liter.

The mean serves as the center point of our distribution. In a normal distribution, which we assume for this exercise, the mean divides the dataset such that half of the values are below it, and half are above it. It essentially balances our dataset like a fulcrum in a seesaw, providing a reference for understanding the spread of our data around this central value.

Understanding the mean helps in making predictions and interpreting data trends. For example, if a single water source has 50 micrograms per liter, it's higher than the average, indicating it's more contaminated than a typical water source in this dataset.

Standard Deviation

The "standard deviation" is another vital element in understanding data distribution, represented by the symbol \( \sigma \). It gives us an idea of how spread out the data values are from the mean. For our data, the standard deviation is 53 micrograms per liter.

A large standard deviation signifies that the chloroform amounts vary widely from the average, meaning there's more diversity in chloroform levels among different water sources. Conversely, a smaller standard deviation would mean the chloroform levels are more consistent across various sources.

This metric is crucial for understanding the spread of the data and predicting variability in measurements. For instance, in a normal distribution:

• About 68% of values lie within one standard deviation from the mean.
• Approximately 95% are within two standard deviations.
• Nearly 99.7% fall within three standard deviations.

Such insights are instrumental in assessing risk, making decisions, and grasping the overall consistency of our data.

Bell Curve

The "bell curve" is another common term for the graph of a normal distribution. It's called so because of its bell-shape: wide in the middle and tapering off towards the ends. This curve represents how data points are distributed across values.

In a bell curve, the highest point of the curve is at the mean, showing that this is where most of the data points lie. As we move away from the mean, the likelihood of observing a data point decreases. This tapering happens because fewer observations stray far from the average.

The bell curve is symmetric, meaning the spread is even on both sides from the mean. When dealing with a bell curve, standard deviation helps us understand how tightly or loosely the data points cluster around the mean.

• One standard deviation from the mean covers the curve's most significant portion, capturing the bulk of data points.
• More extreme values, or "outliers," fall further to the left or right, which the tails of the bell curve represent.

This concept is at the heart of many statistical analyses and helps in understanding how typical or atypical observations are within any dataset.