Problem 17 - Algebra 2

Question

For Exercises \(17-19,\) use the following information. The sizes of CDs made by a company are normally distributed with a standard deviation of 1 millimeter. The CDs are supposed to be 120 millimeters in diameter, and they are made for drives 122 millimeters wide. What percent of the CDs would you expect to be greater than 120 millimeters?

Step-by-Step Solution

Answer

50% of CDs are expected to be greater than 120 mm in diameter.

Step 1: Understand the Normal Distribution

The problem states that the CD sizes are normally distributed with a mean (expected diameter) of 120 mm and a standard deviation of 1 mm. This means that the distribution of CD sizes follows a bell-shaped curve centered at 120 mm, with most CDs being close to this size.

Step 2: Define the Z-Score

To find out what percentage of CDs are greater than 120 mm, we need to calculate the Z-score. The Z-score tells us how many standard deviations a value is from the mean. The formula for the Z-score is: \( Z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value of interest, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In this case, because we are interested in CDs greater than 120 mm, we set \( X = 120 \), \( \mu = 120 \), and \( \sigma = 1 \).

Step 3: Calculate the Z-Score

Plug the values into the Z-score formula: \( Z = \frac{(120 - 120)}{1} = 0 \). This means that CDs with a size of 120 mm are exactly the mean, or the center, of the distribution.

Step 4: Use the Z-Score to Find Probability

For a standard normal distribution (mean = 0), a Z-score of 0 corresponds to the 50th percentile. Since we want the percentage of CDs greater than 120 mm, we look at the upper half of the distribution. The total probability for CDs greater than 120 mm is the area to the right of a Z-score of 0, which is 50%.

Key Concepts

Understanding the Standard Deviation

The standard deviation is a fundamental concept in statistics that measures the amount of variation or dispersion in a set of values. For our CD example, a standard deviation of 1 mm means that the sizes typically deviate from the average by about 1 mm.

In a normal distribution, approximately 68% of the data lies within one standard deviation of the mean. This predictable spread helps us understand and make predictions about our data set:

• 68% within one standard deviation
• 95% within two standard deviations
• 99.7% within three standard deviations

Knowing this, if the CDs are supposed to be 120 mm on average, having a standard deviation of 1 mm lets us assume that most CDs are quite close to this ideal size.

Z-score: The Standardization Tool

A Z-score is a statistic that tells us how many standard deviations a data point is from the mean. It's crucial for comparing data points from different distributions. In the context of the CD sizes:

• Any size exactly at the mean has a Z-score of 0.
• Positive Z-scores indicate values above the mean.
• Negative Z-scores indicate values below the mean.

To calculate the Z-score, we use the formula: \[ Z = \frac{(X - \mu)}{\sigma} \]where:

• \( X \) is the data point you're evaluating (in this case, 120 mm).
• \( \mu \) is the average or mean of the set (120 mm for the CDs).
• \( \sigma \) is the standard deviation (1 mm for these CDs).

In this problem, the Z-score of 0 means that a CD size of 120 mm is exactly at the mean, making it a central point in our distribution.

Demystifying Percentiles

Percentiles offer a way to understand the position of a particular score in relation to the rest of the data. When you know a Z-score, you can easily find a percentile using a standard normal distribution table.

• The 50th percentile, for instance, represents the median, where half the values are above and half are below.
• To find the proportion of CDs larger than 120 mm, starting with a Z-score of 0, you look at the proportion of data to its right.

Given a Z-score of 0 corresponds to the 50th percentile, we understand that 50% of CDs are larger than 120 mm because we're considering the right side of the bell curve.

The Bell-Shaped Curve of Normal Distribution

When data is distributed normally, the "bell-shaped curve" metaphor becomes very helpful. The normal distribution visually appears as a symmetrical curve centered around the mean, where:

• The peak of the curve shows the mean of the data set, such as the CD diameter of 120 mm.
• The spread of the curve is determined by the standard deviation; a more spread-out curve indicates a larger standard deviation.
• The tails of the curve fall off quickly, representing the rare occurrences far from the mean.

Recognizing a normal distribution's bell-shaped form helps in predicting probabilities and visualizing data behaviors. The normal distribution's bell curve dictates that most data falls near the mean, while the probability of extreme values decreases as you move farther from the center.