Problem 23 - 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 more than 210 days?

Step-by-Step Solution

Answer

Approximately 15.87% of the products last more than 210 days.

Step 1: Identify Key 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 want to find the percentage of products that last more than 210 days.

Step 2: Calculate the Z-score

To find out how many products last more than 210 days, we first calculate the Z-score, which tells us how many standard deviations away 210 days is from the mean. The formula for finding the Z-score is: \[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is 210 days, \(\mu\) is 180 days, and \(\sigma\) is 30 days.Plug in the values:\[ Z = \frac{210 - 180}{30} = \frac{30}{30} = 1 \]

Step 3: Use the Z-table

Use the Z-table to determine the probability of a Z-score less than 1. A Z-score of 1 corresponds to approximately 0.8413, meaning that 84.13% of the distribution lies below 210 days.

Step 4: Calculate the Complement

Since we need the percentage of products lasting more than 210 days, we calculate the complement of the probability. Subtract the cumulative probability found in the Z-table from 1:\[ 1 - 0.8413 = 0.1587 \]This means 15.87% of the products last more than 210 days.

Key Concepts

Mean

In statistics, the mean is a measure of central tendency, often referred to as the average. It provides us with a single value that summarizes the center of a data set. In the context of a normal distribution, the mean is denoted by \( \mu \) and represents the peak of the bell curve.

To calculate the mean of a data set, add up all the values and divide by the number of observations. For example, if the shelf life of a snack chip is normally distributed with a mean of 180 days, this number tells us that most chips have a shelf life centred around 180 days.

• The mean is a very informative number since it helps us understand the general performance of a data set.
• In the normal distribution, the mean is in the middle, marking the point where half of the data lies.

Standard Deviation

Standard deviation, represented by \( \sigma \), quantifies the amount of variation or dispersion in a set of data values. In simpler terms, it explains how spread out the numbers in a data set are around the mean.

In any normal distribution, approximately 68% of the data falls within one standard deviation of the mean. If the standard deviation is smaller, the data points are closer to the mean, forming a narrow and tall curve. On the other hand, a larger standard deviation leads to a flatter and wider curve.

• Understanding standard deviation is essential because it shows us how reliable or variable our data is.
• A standard deviation of 30 days, as in our snack chip example, suggests that most chips have a shelf life close to 180 days but can vary by about 30 days.

Z-score

The Z-score is a statistical measurement that describes a value's position in relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean.

The Z-score is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the value being evaluated (like 210 days for our snack chips), \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

• A Z-score of 1 means the value is one standard deviation above the mean, while a Z-score of -1 means it's one standard deviation below.
• Z-scores allow us to compare data from different normal distributions. In our example, a Z-score of 1 tells us that 210 days is one standard deviation above the mean shelf life of 180 days.

Cumulative Probability

Cumulative probability refers to the probability that a normally distributed random variable is less than or equal to a particular value. It is often determined using a Z-table, which shows the probability of obtaining a Z-score less than a given number.

In our example, after calculating a Z-score of 1, we found that about 84.13% of snack chips last 210 days or less. Cumulative probability helps us understand how a specific observation relates to the rest of the data by determining what percentage of observations fall below it.

• To find how many chips last longer than 210 days, we use the complement rule: simply subtract the cumulative probability from 1.
• This staple concept in statistics, with a cumulative probability of 0.8413, means 15.87% of the snacks will last beyond 210 days.