Problem 22 - Algebra 2

Question

FOOD For Exercises \(20-23\) , use the following information. The shelf life of a particular snack chip is normally distributed with a mean of 180 days and a standard deviation of 30 days. About what percent of the products last less than 90 days?

Step-by-Step Solution

Answer

Approximately 0.13% of products last less than 90 days.

Step 1: Identify the parameters

The problem states that the shelf life of the snack chip follows a normal distribution with a mean (\( \mu \)) of 180 days and a standard deviation (\( \sigma \)) of 30 days. We need to find the percentage of chips that last less than 90 days.

Step 2: Calculate the Z-score

The Z-score formula is \( z = \frac{x - \mu}{\sigma} \). Here, \( x = 90 \), \( \mu = 180 \), and \( \sigma = 30 \). Substitute these values into the formula to find the Z-score. \[ z = \frac{90 - 180}{30} = \frac{-90}{30} = -3 \]

Step 3: Find the probability using the Z-score

Using a standard normal distribution table, locate the Z-score of -3. The Z-score table tells us the probability of a value being less than the specified Z-score. For \( z = -3 \), this probability is approximately 0.0013.

Step 4: Convert probability to percentage

The probability found from the Z-score table is 0.0013. To convert this into a percentage, multiply by 100. \[ 0.0013 \times 100 = 0.13\% \]

Step 5: Conclusion

Thus, about 0.13% of the products last less than 90 days. This is a very small percentage indicating that very few products have such a short shelf life.

Key Concepts

Z-score

The Z-score is a crucial concept when dealing with normal distributions. It tells us how many standard deviations a specific score (or value) is from the mean of the distribution. In simple terms, it helps us understand where a particular value stands relative to the typical values in a dataset.

To calculate the Z-score, you use the formula:\[z = \frac{x - \mu}{\sigma}\]Where:

• \(x\) is the value you're evaluating (in our exercise, this is 90 days).
• \(\mu\) is the mean of the distribution (here, it is 180 days).
• \(\sigma\) is the standard deviation (in this case, 30 days).

Once you've calculated the Z-score, you use it to determine how likely or unlikely a result is. In our example, the Z-score of -3 means the value (shelf life of 90 days) is 3 standard deviations below the mean, indicating it is significantly lower than the average.

standard deviation

Standard deviation is a measure of dispersion in a set of data. It reflects how much the values in a dataset deviate from the mean. If the standard deviation is small, the data points are close to the mean. If it's large, they're spread out over a wider range of values.

In our example about snack chips, a standard deviation of 30 days suggests that most snack chip shelf lives will vary by about 30 days from the average of 180 days. This measure helps us understand the variability and reliability of the shelf life. Standard deviation is a key element in various statistical analyses and real-world applications, such as quality control in manufacturing. It helps businesses and researchers assess consistency and make informed decisions.

mean

The mean, often referred to as the average, is a fundamental statistic used to describe the central tendency of a data set. It is calculated by summing all the individual data points and then dividing by the number of points.

For the shelf life of the snack chips, the mean is given as 180 days. This value serves as the central point around which the other shelf life values are distributed. Knowing the mean helps us understand what is typical or expected in the dataset. The mean is widely used alongside other measures like median and mode to provide a comprehensive overview of data. While it is sensitive to extreme values, in normally distributed datasets like this one, the mean is usually a reliable indicator of the general trend.