Problem 5 - 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 between \(90,000\) and \(110,000\) miles?

Step-by-Step Solution

Answer

About 13,652 batteries will last between 90,000 and 110,000 miles.

Step 1: Identify Given Information

The problem states that the useful life of the car battery is normally distributed with a mean \( \mu = 100,000 \) miles and a standard deviation \( \sigma = 10,000 \) miles. We need to find out how many batteries will last between \( 90,000 \) miles and \( 110,000 \) miles out of 20,000 produced.

Step 2: Convert to Standard Normal Distribution

To find the probability of batteries lasting between \( 90,000 \) and \( 110,000 \) miles, we convert these to standard scores (z-scores). The z-score formula is \( z = \frac{X - \mu}{\sigma} \).- For 90,000 miles: \( z_1 = \frac{90,000 - 100,000}{10,000} = -1 \)- For 110,000 miles: \( z_2 = \frac{110,000 - 100,000}{10,000} = 1 \)

Step 3: Use Standard Normal Distribution Table

Using a z-table, we find the probability for each z-score.- For \( z = -1 \), the cumulative probability is approximately \(0.1587\).- For \( z = 1 \), the cumulative probability is approximately \(0.8413\).The probability of a battery lasting between \( 90,000 \) and \( 110,000 \) miles is the difference: \( 0.8413 - 0.1587 = 0.6826 \).

Step 4: Calculate Number of Batteries

To find the approximate number of batteries, multiply the total number of batteries by the probability calculated.Number of batteries = \( 20,000 \times 0.6826 = 13,652 \) batteries.

Key Concepts

Mean and Standard Deviation

Understanding mean and standard deviation is crucial when dealing with normal distributions, such as the lifespan of car batteries in this exercise. The **mean** refers to the average value of a dataset, which acts as the 'center' of the distribution. In the car battery example, this is given as 100,000 miles.
On the other hand, the **standard deviation** indicates how spread out the values are around the mean. A smaller standard deviation means data points are close to the mean, while a larger one indicates more spread. Here, the standard deviation is 10,000 miles. Knowing the mean and standard deviation helps assess how typical or unusual a given data point might be.
Recognizing these elements is key because they determine the shape and spread of the normal distribution curve, thus allowing us to calculate probabilities of different outcomes.

Z-Score

The **z-score** is a statistical measurement that describes a value's relation to the mean of a group of values. It is calculated by taking the difference between a particular value and the mean, then dividing by the standard deviation. Mathematically, it is expressed as:

• \( z = \frac{X - \mu}{\sigma} \)

where:

• \( X \) is the value being evaluated.
• \( \mu \) is the mean of the dataset.
• \( \sigma \) is the standard deviation.

This calculation converts real-world figures into a standardized form, making it easier to compare values across different datasets. In the problem, z-scores help find how far values like 90,000 and 110,000 miles are from the mean lifespan of 100,000 miles in standard deviations, resulting in z-scores of -1 and 1 respectively.

Probability Calculation

Probability calculations using z-scores provide insights into the likelihood of observed values within a normal distribution. In our car battery example, once z-scores are determined, they help find the probabilities or the likelihood of the battery life falling between two values—here, between 90,000 and 110,000 miles.
To find this probability, we determine the cumulative probability for each z-score using a z-table. For the z-score of -1, the cumulative probability is approximately 0.1587, and for the z-score of 1, it is approximately 0.8413. The difference between these probabilities gives the probability that the battery life is between 90,000 and 110,000 miles:

• Probability = 0.8413 - 0.1587 = 0.6826

This value indicates that about 68.26% of batteries are likely to last between these two values.

Standard Normal Distribution Table

The **Standard Normal Distribution Table** is an invaluable tool in probability calculation involving normal distributions. It presents cumulative probabilities corresponding to particular z-scores. Essentially, it tells us how much of the distribution falls to the left of a given z-score, thereby supplying probabilities associated with specific values.
In our exercise, using the z-table we found cumulative probabilities for z-scores -1 and 1. With these values, it becomes easier to calculate the probability of a value falling between two points, such as between 90,000 and 110,000 miles in the life of the batteries. By understanding and utilizing the standard normal distribution table, predictions on real-world data sets can be made more accurately, thereby offering significant practical utility.