Problem 20 - 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 between 150 and 210 days?

Step-by-Step Solution

Answer

About 68.26% of the products last between 150 and 210 days.

Step 1: Understanding the Problem

We are given that the shelf life of a snack chip is normally distributed with a mean, \( \mu = 180 \) days, and a standard deviation, \( \sigma = 30 \) days. We are to find the percentage of products that last between 150 and 210 days.

Step 2: Converting to Z-scores

To find the percentage of the products between 150 and 210 days, we first convert these values to Z-scores using the formula \( Z = \frac{X - \mu}{\sigma} \). For 150 days, \( Z_{150} = \frac{150 - 180}{30} = -1 \). For 210 days, \( Z_{210} = \frac{210 - 180}{30} = 1 \).

Step 3: Using the Standard Normal Distribution

Now we look up the Z-scores in the standard normal distribution table (or use a calculator) to find the corresponding probabilities. \( P(Z < -1) \) is approximately 0.1587, and \( P(Z < 1) \) is approximately 0.8413. These give the cumulative probabilities from the left.

Step 4: Finding the Probability Between Z-scores

To find the probability between 150 and 210 days, we calculate \( P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) \). Substituting the values, we get \( P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826 \).

Step 5: Converting Probability to Percentage

The probability 0.6826 corresponds to 68.26% when converted to a percentage. Therefore, about 68.26% of the snack chips have a shelf life between 150 and 210 days.

Key Concepts

Understanding Z-scores

The concept of a z-score is pivotal in understanding the normal distribution. A z-score helps us identify how many standard deviations an element is from the mean. In the context of our exercise, to find out about the chip's shelf life distribution, we calculated the z-scores for 150 days and 210 days.

• The formula for calculating a z-score is: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• For 150 days: \( Z_{150} = \frac{150 - 180}{30} = -1 \).
• For 210 days: \( Z_{210} = \frac{210 - 180}{30} = 1 \).

These scores show us how far each value is from the mean in terms of standard deviations. Z-scores allow us to use a standard normal distribution table to find probabilities.

Role of Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In a normal distribution, it tells us how much the data differs from the mean (average). The wider the spread (greater the standard deviation), the more varied the data points are.

• In our snack chip example, the standard deviation is 30 days.
• This means the shelf life of most snack chips deviates 30 days from the mean of 180 days, either positively or negatively.

Standard deviation is a critical component when calculating z-scores, as it gives us a scale to measure how far a given value is from the mean, ultimately helping us determine where a value falls within the distribution.

Exploring Cumulative Probability

Cumulative probability helps us find the probability that a variable falls within a particular range in a distribution. When dealing with normal distributions, cumulative probability allows us to see what portion of the data falls below a certain value.

• The standard normal distribution table, or calculator, gives us this probability for any given z-score.
• In the exercise, probabilities associated with \( Z < -1 \) and \( Z < 1 \) were found to be 0.1587 and 0.8413, respectively.

By taking the difference between these cumulative probabilities, \( P(Z < 1) - P(Z < -1) \), we determine the probability that a value falls between these two z-scores. This method is crucial in finding out what percent of the products have a shelf life within the given range.

Calculating Percentages in Statistics

Percentage calculation in statistics is used to interpret probabilities in a more intuitive way. Once we have the probability as a decimal, converting it to a percentage clarifies what portion of the data set our interest covers.

• To convert a probability to a percentage, simply multiply by 100.
• In the exercise, the probability of 0.6826 translates to 68.26%.

By reporting the result as a percentage, we can easily communicate and understand what proportion of the snack chips fall between the specified shelf life durations.