Problem 25 - Basic Statistics for Business and Economics

Question

The amounts dispensed by a cola machine follow the normal distribution with a mean of 7 ounces and a standard deviation of 0.10 ounces per cüp. How much cola is dispensed in the largest 1 percent of the cups?

Step-by-Step Solution

Answer

About 7.233 ounces are dispensed in the largest 1% of cups.

Step 1: Understanding the Problem

The task is to find the amount of cola dispensed that falls into the largest 1% of the distribution. This refers to the value at the 99th percentile of the normal distribution with mean (\( \mu \)) = 7 ounces and standard deviation (\( \sigma \)) = 0.10 ounces.

Step 2: Locate the Z-score

To find the amount dispensed in the largest 1%, we need to locate the Z-score that corresponds to the 99th percentile of the standard normal distribution. By using a standard normal distribution table, or a calculator, we find that Z ≈ 2.33.

Step 3: Apply the Z-score Formula

Once we have the Z-score, we can use the formula for the value in a normal distribution: \[ X = \mu + Z \cdot \sigma \]Substitute \( \mu = 7 \) ounces, \( Z = 2.33 \), and \( \sigma = 0.10 \) ounces into the formula.

Step 4: Calculate the Dispensed Amount

Substitute the known values into the formula: \[ X = 7 + 2.33 \cdot 0.10 \]Calculate:\[ X = 7 + 0.233 = 7.233 \] Thus, the amount of cola dispensed in the largest 1% of the cups is approximately 7.233 ounces.

Key Concepts

Percentiles

In statistics, percentiles are a way to understand how a particular value compares to the rest of a dataset. When you look at a dataset as a whole, percentiles will help you determine how far or how high a certain value falls compared to others. For instance, the 50th percentile is also known as the median, meaning half of the data values fall below it.

• The 99th percentile indicates that 99% of the data values are below this point, and just 1% are above.
• Percentiles readily help in identifying edge cases or high flyers in your data.

Once you determine the percentile you are interested in (like the 99th percentile for the cola machine problem), you will often use a Z-score to find the exact value at that point in a normal distribution.

Z-score

A Z-score is a measure of how many standard deviations a data point is away from the mean. In the context of a standard normal distribution, it helps you quantify the distance a certain value lies from the average or mean value. Z-scores play an essential role in various fields because they allow for comparison across different datasets, even if those datasets have different means and standard deviations.

To determine the Z-score corresponding to a specific percentile in a normal distribution:

• Standard normal distribution tables are used to find Z-scores for common percentile values. For example, a Z-score of 2.33 corresponds roughly to the 99th percentile.
• This means that about 1% of the distribution will have higher values than what corresponds to a Z-score of 2.33.

Mean

The mean is the average of a set of data points and is one of the central concepts in statistics. It provides a simple measure to understand where the center of the data distribution lies.
For example, if you add up all the values in a data set and then divide by the number of values, you've found the mean. This is incredibly useful in creating a basic understanding of the dataset's tendencies.

• The mean is denoted by the Greek letter \( \mu \) in statistical equations.
• A key property of the mean in a normal distribution is that it serves as the point around which the values are symmetrically distributed.

In the cola machine example, knowing that the mean is 7 ounces helps you understand the base level of cola dispensed by the machine.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are around the mean. It is crucial for assessing the variability within a dataset.
A dataset with a high standard deviation means that the data points are spread out over a wider range of values, while a low standard deviation signifies that the values are closely clustered around the mean.

• In statistics, the standard deviation is denoted by the Greek letter \( \sigma \).
• An important property of the normal distribution is that approximately 68% of the data lies within 1 standard deviation of the mean, about 95% within 2 standard deviations, and roughly 99.7% within 3 standard deviations.

For the cola machine, with a standard deviation of 0.10 ounces, we understand that most cups will be close to the 7 ounces mean, but a few can deviate slightly, causing very small usual variance in amounts dispensed.