91Ó°ÊÓ

E. Coli in Swimming Areas. To investigate water quality, the Columbus Dispatch took water specimens at 16 Ohio State Park swimming areas in central Ohio. Those specimens were taken to laboratories and tested for \(E\). coli, which are bacteria that can cause serious gastrointestinal problems. For reference, if a 100-milliliter specimen (about \(3.3\) ounces) of water contains more than 130 E. coli bacteria, it is considered unsafe. Here are the \(E\). coli levels per 100 milliliters found by the laboratories:2 ECOLI \(\begin{array}{rrrrrrrr}291.0 & 10.9 & 47.0 & 86.0 & 44.0 & 18.9 & 1.0 & 50.0 \\\ 190.4 & 45.7 & 28.5 & 18.9 & 16.0 & 34.0 & 8.6 & 9.6\end{array}\) Find the mean \(E\). coli level. How many of the lakes have \(E\). coli levels greater than the mean? What feature of the data explains the fact that the mean is greater than most of the observations?

Step by step solution

01

Sum of E. Coli Levels

To find the mean, we first need to sum up all the provided E. coli levels. Add each number together: \(291.0 + 10.9 + 47.0 + 86.0 + 44.0 + 18.9 + 1.0 + 50.0 + 190.4 + 45.7 + 28.5 + 18.9 + 16.0 + 34.0 + 8.6 + 9.6 = 900.5\)

02

Calculate the Mean

Now that we have the total sum of the E. coli levels, divide it by the number of observations (16) to get the mean. Thus, the mean is \( \frac{900.5}{16} = 56.28125\).

03

Count Observations Greater Than the Mean

Count how many E. coli levels are greater than the mean of 56.28125. These are: \(291.0, 86.0, 190.4\). Thus, there are 3 observations greater than the mean.

04

Discuss Data Features Affecting the Mean

Note that the mean is greater than most observations due to a few very high values (e.g., 291.0, 190.4) which skew the average higher, demonstrating the effect of skewness on the mean.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

When we talk about calculating the mean, we are finding the average value of a set of numbers. This is a fundamental aspect of descriptive statistics. To compute the mean, follow these simple steps:

1. **Add up all the values**: In the given problem, we summed all the E. coli levels to get a total of 900.5. This is an essential step because you need the grand total to calculate the mean.

2. **Divide by the number of observations**: Take the total sum and divide it by how many numbers there are. Here, we divided 900.5 by 16, which is the number of lakes. This gave us a mean of approximately 56.28.

Understanding mean calculation is crucial since it gives a simple representation of the central value of the data set. It's often used to determine if a particular environment, like water in the lakes, meets a standard set for safety levels.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information and support decision making. In the case of E. coli levels, data analysis helps us understand the water quality in each lake.

Here's how data analysis can be broken down for this example:

• **Identify Patterns**: By calculating the mean, we see a central trend or pattern in E. coli concentrations across different swimming areas. This helps us assess whether many locations exceed safety limits set by health authorities.
• **Compare Observations**: By identifying which specific lakes have levels higher than the overall mean, we can focus on problem areas needing more attention.
• **Assess Safety**: The analysis provides insights into how safe each lake is based on its E. coli measurement against the mean. This is crucial for public health and safety.

By effectively analyzing the data, we can identify and prioritize areas that may require immediate action, like water treatment or public advisories.

Skewed Data

Skewed data refers to when the data is not symmetrically distributed around the mean. In the exercise, a few extremely high E. coli levels skew our dataset, making the mean greater than most individual observations.

Key features of skewed data include:

• **Influence of Extreme Values**: In our exercise, high values like 291.0 and 190.4 significantly raise the overall mean, despite many other values being lower. This is common in positively skewed data where the "tail" of the distribution stretches toward higher values.
• **Impact on Mean**: Skewed data can cause the mean to be different from the median (the middle value). Thus, the mean might not always be the best representation of central tendency in skewed distributions.
• **Understanding Data Spread**: Recognizing skewness helps us understand the spread and tendencies of data. It also signals the potential need for different statistical approaches or transformations in analysis.

By understanding skewness, statisticians and data analysts can better interpret the data and make informed conclusions about the dataset. This understanding helps to ensure actions taken are based on accurate interpretations of the data.