Question

Suppose the numbers of a particular type of bacteria in samples of 1 milliliter \((\mathrm{ml})\) of drinking water tend to be approximately normally distributed, with a mean of 85 and a standard deviation of \(9 .\) What is the probability that a given 1 -ml sample will contain more than 100 bacteria?

Short answer

Answer: The probability that a 1-ml sample of drinking water contains more than 100 bacteria is approximately 0.0475 or 4.75%.

Step-by-step solution

Write down the known values

We know the following information about the distribution of bacteria in the drinking water samples: - Mean (μ) = 85 - Standard Deviation (σ) = 9 - We want to find the probability of a sample containing more than 100 bacteria.

Find the Z-score of the desired value

To find the probability associated with a value, we first need to find its Z-score. A Z-score tells us how many standard deviations a value is from the mean. We can calculate the Z-score using the following formula: \(Z = \frac{X - μ}{σ}\) where X is the value we are interested in (100 in this case), μ and σ are the mean and standard deviation of the distribution, respectively. So, plugging the values into the formula, we get: \(Z = \frac{100 - 85}{9} = \frac{15}{9} = 1.67\) The Z-score for 100 bacteria is 1.67.

Find the probability using the Z-score

Now that we have the Z-score, we can use a Z-table to find the probability of a 1-ml sample containing more than 100 bacteria. A Z-table tells us the probability that a value from a standard normal distribution is less than or equal to a given Z-score. We will be using a Z-table for positive Z-scores. Since we want to find the probability of more than 100 bacteria, we need to find the area to the right of the Z-score (1.67) in the Z-table. The area to the left of the Z-score corresponds to the probability that a value is less than or equal to the given Z-score. So we find the area to the left of the Z-score in the Z-table and subtract it from 1 to get the area to the right. Looking up 1.67 in the Z-table, we find a probability of 0.9525. Now, since we want the probability that a 1-ml sample will have more than 100 bacteria, we need the area to the right of the Z-score. This is calculated as: \(P(X > 100) = 1 - P(X \leq 100) = 1 - 0.9525 = 0.0475\)

State the final answer

The probability that a given 1-ml sample of drinking water will contain more than 100 bacteria is approximately 0.0475 or 4.75%.

Key concepts

Normal Distribution

Understanding the normal distribution is fundamental in statistics, especially when it comes to analyzing data that tends to cluster around a central value. The normal distribution, often referred to as the Gaussian distribution, is a bell-shaped curve where the bulk of the values lie near the mean. In the context of our exercise, the number of bacteria in samples of drinking water is assumed to follow this pattern.

This type of distribution is symmetric, meaning the left and right sides of the curve are mirror images. Additionally, the mean, median, and mode of a perfectly normal distribution are equal. When the distribution of data is normal, we can effectively use standard deviation and Z-scores to calculate probabilities for specific ranges or values of interest.

Z-score Calculation

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It's a way of standardizing scores on the same scale to easily compare them. To calculate a Z-score, we use the formula:
\[Z = \frac{X - \mu}{\sigma}\]
where \(X\) is the value in question, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. The resulting Z-score tells us how far and in what direction the value deviates from the mean. A high Z-score indicates a value far above the mean, while a negative Z-score signifies a value far below the mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a larger range. In our exercise, we're given a standard deviation of 9, which tells us on average, the number of bacteria per 1 ml sample varies by 9 from the mean.

It's important to know that standard deviation can be greatly influenced by outliers. In datasets with extreme values, standard deviation may not provide an accurate sense of variability. However, for normally distributed data, as we have in the exercise, the standard deviation is a very helpful tool to assess the consistency of the data.

Probability Tables

Probability tables, such as Z-tables, are used to find the probability that a statistic, like the Z-score, is less than (or sometimes greater than) a certain value. These tables are built from the standard normal distribution, which has a mean of 0 and a standard deviation of 1. By looking up the Z-score in the Z-table, we can find the cumulative probability associated with that Z-score.

For values larger than the mean, we find the area to the left as given in the table and subtract it from one to find the area to the right. This area represents the probability we are seeking, which is the probability that the value falls in the upper tail of the distribution. In practical terms, as seen in the exercise, Z-tables help us translate Z-scores into actual probabilities, answering questions like the likelihood of finding a certain number of bacteria in a water sample.