Problem 29 - Basic Statistics for Business and Economics

Question

The amount of cola in a 12 -ounce can is uniformly distributed between 11.96 ounces and 12.05 ounces. a. What is the mean amount per can? b. What is the standard deviation amount per can? c. What is the probability of selecting a can of cola and finding it has less than 12 ounces? d. What is the probability of selecting a can of cola and finding it has more than 11.98 ounces? e. What is the probability of selecting a can of cola and finding it has more than 11.00 ounces?

Step-by-Step Solution

Answer

a. Mean: 12.005 oz b. Std Dev: 0.02598 oz c. P(X<12.): 0.4444 d. P(X>11.98): 0.7778 e. P(X>11.00): 1

Step 1: Identify the Distribution

The problem indicates that the amount of cola in the can is uniformly distributed between 11.96 ounces and 12.05 ounces. This is a uniform distribution with parameters a = 11.96 and b = 12.05.

Step 2: Calculate the Mean

The mean of a uniform distribution is calculated as \( \frac{a + b}{2} \). Therefore, the mean amount per can is \( \frac{11.96 + 12.05}{2} = 12.005 \) ounces.

Step 3: Calculate the Standard Deviation

The standard deviation for a uniform distribution is \( \frac{b-a}{\sqrt{12}} \). Substituting the values, we have \( \frac{12.05 - 11.96}{\sqrt{12}} = \frac{0.09}{\sqrt{12}} \approx 0.02598 \) ounces.

Step 4: Calculate Probability of Less than 12 Ounces

To find the probability of a can containing less than 12 ounces, calculate the proportion of the interval. Since 12 ounces is within the range, the probability is \( \frac{12 - 11.96}{12.05 - 11.96} = \frac{0.04}{0.09} \approx 0.4444 \).

Step 5: Calculate Probability of More than 11.98 Ounces

Calculate the probability of a can containing more than 11.98 ounces as \( \frac{12.05 - 11.98}{12.05 - 11.96} = \frac{0.07}{0.09} \approx 0.7778 \).

Step 6: Calculate Probability of More than 11.00 Ounces

Since 11.00 ounces is outside the range to the left, the entire distribution is more than 11.00 ounces. Thus, the probability is 1.

Key Concepts

Mean Calculation

The average amount of a uniformly distributed can of cola can be found by calculating the mean. In a uniform distribution, each outcome is equally likely within a defined range, which makes calculating the mean straightforward. The formula for finding the mean in a uniform distribution is given by \[ \text{Mean} = \frac{a + b}{2} \] where \( a \) is the lower bound and \( b \) is the upper bound of the distribution. For the cola can example, this means identifying \( a = 11.96 \) ounces and \( b = 12.05 \) ounces. Plugging these values into the formula, the mean calculates to \[ \text{Mean} = \frac{11.96 + 12.05}{2} = 12.005 \text{ ounces} \]. This mean represents the expected average volume in each can when you consider all the cans together.

Standard Deviation

The standard deviation in a uniform distribution measures the dispersion or spread of data within the distribution’s range. It tells us how much values typically deviate from the mean. The calculation of standard deviation for a uniform distribution is slightly different from normal distributions and is calculated as follows: \[ \text{Standard Deviation} = \frac{b-a}{\sqrt{12}} \] For the cola in this exercise, where \( b = 12.05 \) and \( a = 11.96 \), you can calculate: \[ \text{Standard Deviation} = \frac{12.05 - 11.96}{\sqrt{12}} = \frac{0.09}{\sqrt{12}} \approx 0.02598 \text{ ounces} \]. This result implies that the amount of cola per can typically deviates from the mean by about 0.026 ounces.

Probability of Uniform Distribution

Understanding the probability of an event within a uniform distribution involves evaluating the portion of the total range that the event represents.
To find the probability of selecting a can that has less than 12 ounces, consider the range from 11.96 to 12.05 ounces. The probability is calculated as the length of the segment up to 12 ounces over the total length of the distribution. Thus, \[ \text{Probability} = \frac{12 - 11.96}{12.05 - 11.96} = \frac{0.04}{0.09} \approx 0.4444 \].
For a can containing more than 11.98 ounces, the relevant segment stretches from 11.98 to 12.05. The probability here is \[ \frac{12.05 - 11.98}{12.05 - 11.96} = \frac{0.07}{0.09} \approx 0.7778 \].
Lastly, finding a can with more than 11.00 ounces covers the entire distribution range, resulting in a probability of 1, as every possible outcome falls within the specified range.

Statistical Problem Solving

Statistical problem solving using uniform distribution involves applying straightforward formulas to find mean and standard deviation, and using simple proportions to calculate probabilities. It’s essential to first identify parameters such as \( a \) and \( b \) (the range), and then execute calculations logically.

• Start by clearly determining your range of data (from \( a \) to \( b \)).
• Use the mean calculation for insights into expected values.
• Apply the standard deviation formula to understand variability.
• Determine probabilities by calculating the proportion of the range that satisfies your condition.

These steps provide a systematic way to solve statistical problems involving uniform distributions. They help ensure accuracy and aid in understanding data behavior within its range.