Question

Packages of food whose average weight is 16 ounces with a standard deviation of 0.6 ounces are shipped in boxes of 24 packages. If the package weights are approximately normally distributed, what is the probability that a box of 24 packages will weigh more than 392 ounces \((24.5\) pounds \() ?\)

Short answer

The probability that a box of 24 packages will weigh more than 392 ounces is approximately 0.34%.

Step-by-step solution

Find the mean and standard deviation of the total weight

We are given the average weight of a single package (16 ounces) and the standard deviation (0.6 ounces). Let's call the average weight of a single package \(\mu\). Thus, \(\mu = 16\) ounces. Let's call the standard deviation of a single package \(\sigma\). Thus, \(\sigma = 0.6\) ounces. Now, we need to find the mean and standard deviation of the total weight of 24 packages. This is a sum of 24 independent random variables, so we can use the following formulas: Mean of the total weight of 24 packages: \(\mu_{total} = 24\mu\) Standard deviation of the total weight of 24 packages: \(\sigma_{total} = \sqrt{24} \cdot \sigma\) Now, calculate \(\mu_{total}\) and \(\sigma_{total}\).

Calculate the mean and standard deviation of the total weight

Using the formulas from Step 1, calculate the mean and standard deviation of the total weight of 24 packages: \(\mu_{total} = 24 \cdot 16 = 384\) ounces \(\sigma_{total} = \sqrt{24} \cdot 0.6 = 2.95\) ounces (rounded to two decimal places)

Convert the problem into a standard normal distribution problem

We want to find the probability that a box of 24 packages will weigh more than 392 ounces. Let's call this weight \(w\). We can convert this problem into a standard normal distribution problem by finding the z-score. The z-score is defined as: \(z = \frac{w - \mu_{total}}{\sigma_{total}}\) Now, calculate the z-score for 392 ounces.

Calculate the z-score

Using the formula from Step 3, calculate the z-score for 392 ounces: \(z = \frac{392 - 384}{2.95} = 2.71\) (rounded to two decimal places)

Find the probability using a z-table or a calculator

Now that we have the z-score, we can find the probability that a box of 24 packages will weigh more than 392 ounces using a z-table or a calculator. A z-table gives us the probability that a value from a standard normal distribution is less than the z-score. Since we want the probability that the weight is more than 392 ounces (greater than the z-score), we need to find the complement of the probability given by the z-table: \(P(Z > 2.71) = 1 - P(Z < 2.71)\). Using a z-table or calculator, find \(P(Z < 2.71)\) and then calculate \(P(Z > 2.71)\).

Calculate the probability

Using a z-table or a calculator, we find that \(P(Z < 2.71) \approx 0.9966\). So, the probability that a box of 24 packages will weigh more than 392 ounces is: \(P(Z > 2.71) = 1 - P(Z < 2.71) = 1 - 0.9966 = 0.0034\) Thus, the probability that a box of 24 packages will weigh more than 392 ounces is approximately 0.34%.

Key concepts

Normal Distribution

The normal distribution is a probability distribution that is symmetric about the mean, meaning it forms a bell-shaped curve when plotted on a graph. Most of the data points lie close to the mean, and the probability of values falling further away from the mean decreases as you move outwards. This distribution is crucial in statistics because many natural phenomena follow this pattern. In our exercise, the weights of food packages are assumed to follow a normal distribution. This assumption allows us to make predictions about the probability of certain events, such as the total weight of multiple packages.

Characteristics of a normal distribution include:

• Symmetrical bell shape
• Mean, median, and mode are equal
• Defined by two parameters: mean (\( \mu \)) and standard deviation (\( \sigma \))

This distribution makes it easier to apply statistical methods, like calculating the probability that a set of packages weighs more than a certain amount.

Standard Deviation

Standard deviation is a measure of how spread out numbers are in a data set. It indicates the amount of variability or dispersion from the average (mean). A low standard deviation means that the data points tend to be close to the mean, while a high standard deviation indicates that they are spread out over a wider range of values, which is critical for understanding the reliability of the mean.

In our exercise, the standard deviation of a single package is given as 0.6 ounces. This shows us that the individual package weights don't vary too much from the average weight. When considering multiple packages, like 24 in this scenario, the total weight's standard deviation is calculated using \( \sigma_{total} = \sqrt{n} \cdot \sigma \), where \( n \) is the number of packages. This formula reflects the fact that total weight has more variability than individual package weights but remains manageable because it increases by the square root of the number of packages.

Z-Score

A z-score is a statistical measurement that describes how many standard deviations a data point is from the mean. It helps in understanding the position of a specific value in relation to the distribution of the data set. By converting individual data points into a standard normal distribution, z-scores allow for comparison across different data sets or distributions.

To find the z-score, you use the formula: \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the data point in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In our case, we calculated the z-score for the total weight of 392 ounces of packages. The result, 2.71, tells us how far above the mean this weight is in terms of standard deviations, thus enabling probability calculations using the standard normal distribution.

Mean

The mean, or average, is a central measure in statistics, representing the sum of values divided by the count of those values. It provides a central point around which data values are distributed, and it's a primary measure of central tendency.

In our problem, the mean weight of a single package is 16 ounces. For the total weight of 24 packages, the mean is calculated by multiplying the mean of one package by the number of packages: \( \mu_{total} = 24 \times 16 = 384 \) ounces. This helps determine the expected total weight and serves as a baseline to understand weight variance among the packages. The mean plays a critical role in probability calculations, as it anchors the normal distribution on which z-score evaluations are based.