Problem 7 - Algebra 2

Question

For Exercises \(5-8,\) use the following information. The useful life of a certain car battery is normally distributed with a mean of \(100,000\) miles and a standard deviation of \(10,000\) miles. The company makes \(20,000\) batteries a month. About how many batteries will last less than \(90,000\) miles?

Step-by-Step Solution

Answer

Approximately 3,174 batteries will last less than 90,000 miles.

Step 1: Understand the Problem

We need to find the number of batteries that last less than 90,000 miles. The distribution of battery life is normal with a mean, \( \mu = 100,000 \) miles, and a standard deviation, \( \sigma = 10,000 \) miles. We want to calculate the probability that a battery lasts less than 90,000 miles and then apply it to the number of batteries produced, \( 20,000 \).

Step 2: Convert to Z-score

To find the probability, we first convert the 90,000-mile point to a Z-score using the formula \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value we are interested in. \( Z = \frac{90,000 - 100,000}{10,000} = \frac{-10,000}{10,000} = -1 \).

Step 3: Find the probability from Z-table

Using the Z-score of -1, we refer to the standard normal distribution table (Z-table) to find the probability that a Z-score is less than -1, which is approximately 0.1587 or 15.87%.

Step 4: Calculate the Number of Batteries

Now, multiply the probability of a battery lasting less than 90,000 miles by the total number of batteries produced: \( 0.1587 \times 20,000 = 3,174 \).

Step 5: Verification Step

Recheck the Z-table and calculations to ensure the probability and final calculations are consistent. The probability for Z < -1 should remain approximately 0.1587, leading to about 3,174 batteries.

Key Concepts

Z-score

When talking about normal distribution, the Z-score plays a crucial role. It helps us understand where a specific value stands in relation to the mean of the distribution.
A Z-score is computed using the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

where \( X \) is the value for which we are calculating the Z-score, \( \mu \) is the mean of the distribution and \( \sigma \) is the standard deviation.
The Z-score expresses how many standard deviations a data point is from the mean. In our exercise, the Z-score of -1 means that 90,000 miles is one standard deviation below the mean of 100,000 miles. This Z-score helps us determine the probability of a battery lasting less than 90,000 miles by referencing a Z-table.
Understanding the Z-score is essential since it provides a standardized way to compare different data points within a normally distributed dataset.

Standard Deviation

Standard deviation is a key component in statistical analysis, especially when working with normal distribution.
It measures the amount of variation or dispersion in a set of values.

• A low standard deviation indicates that values tend to be close to the mean.
• A high standard deviation suggests a wide spread around the mean.

For the car battery life example, a standard deviation of 10,000 miles means that most battery lifespans vary by about 10,000 miles from the average of 100,000 miles.
This measurement helps in understanding the reliability and consistency of the car batteries produced. Knowing the standard deviation allows for better predictions about how many batteries will perform significantly different from the average life span.

Probability

Probability in this context is about finding the likelihood that a car battery will last less than a certain number of miles.
Using the Z-score, we referred to the Z-table to find the probability associated with a Z-score of -1.

• For a Z-score of -1, the probability that a battery lasts less than 90,000 miles is about 0.1587 or 15.87%.

This probability tells us that approximately 15.87% of the batteries will have a lifespan below 90,000 miles in a normally distributed production scenario. Understanding probabilities helps in making informed choices about production, inventory management, and quality control assessments.

Statistical Analysis

Statistical analysis is an important process for interpreting data, and with normal distribution, it becomes even more effective.
In our example, statistical analysis involves calculating the Z-score, using standard deviation, and determining probabilities.
These steps help us analyze the performance and expected life of car batteries.

• The Z-score converts a raw data point into a standardized index.
• Standard deviation adds insight into the spread of data points.
• Probability quantifies the likelihood of certain outcomes.

By understanding how all these statistical tools work together, we gain the ability to make data-driven decisions. Such analysis is crucial for manufacturers in predicting product lifespan and ensuring quality consistency across batches.