Problem 31 - Basic Statistics for Business and Economics with Student

Question

Mattel Corporation produces a remote-controlled car that requires three AA batteries. The mean life of these batteries in this product is 35.0 hours. The distribution of the battery lives closely follows the normal probability distribution with a standard deviation of 5.5 hours. As a part of its testing program Sony tests samples of 25 batteries. a. What can you say about the shape of the distribution of the sample mean? b. What is the standard error of the distribution of the sample mean? c. What proportion of the samples will have a mean useful life of more than 36 hours? d. What proportion of the sample will have a mean useful life greater than 34.5 hours? e. What proportion of the sample will have a mean useful life between 34.5 and 36.0 hours?

Step-by-Step Solution

Answer

a) Normal; b) 1.1; c) 18.14%; d) 67.36%; e) 49.22%

Step 1: Shape of the Distribution

The distribution of the sample mean of a normally distributed population is also normal. Therefore, since the battery life follows a normal distribution, the shape of the distribution of the sample mean will also be normal.

Step 2: Calculate the Standard Error

The standard error (SE) of the sample mean is calculated using the formula:\[ SE = \frac{\sigma}{\sqrt{n}} \]where \( \sigma = 5.5 \) hours is the standard deviation of the population and \( n = 25 \) is the sample size. Thus,\[ SE = \frac{5.5}{\sqrt{25}} = \frac{5.5}{5} = 1.1 \text{ hours} \]

Step 3: Proportion for Mean Greater Than 36 Hours

To find the proportion of the samples with a mean greater than 36 hours, calculate the Z-score using:\[ Z = \frac{X - \mu}{SE} \]where \( X = 36 \), \( \mu = 35.0 \), and \( SE = 1.1 \). So,\[ Z = \frac{36 - 35.0}{1.1} = \frac{1}{1.1} \approx 0.91 \]Using a standard normal distribution table, we find the probability corresponding to \( Z = 0.91 \) is approximately 0.8186, so the proportion of samples with a mean over 36 hours is:\[ 1 - 0.8186 = 0.1814 \] or 18.14%.

Step 4: Proportion for Mean Greater Than 34.5 Hours

For \( X = 34.5 \), calculate the Z-score:\[ Z = \frac{34.5 - 35.0}{1.1} = \frac{-0.5}{1.1} \approx -0.45 \]The probability corresponding to \( Z = -0.45 \) is approximately 0.3264. So, the proportion of samples with a mean greater than 34.5 hours is:\[ 1 - 0.3264 = 0.6736 \] or 67.36%.

Step 5: Proportion Between Two Means

To find the proportion of samples with means between 34.5 and 36.0 hours, we use the probabilities from Steps 3 and 4. The proportion of samples between these means is:\[ 0.8186 - 0.3264 = 0.4922 \] or 49.22%.

Key Concepts

Normal Distribution

The normal distribution, often called the bell curve due to its distinctive shape, is a fundamental concept in statistics. It represents how data samples are expected to distribute if influenced only by random factors.
Key features of a normal distribution include:

• Symmetrical shape: The left and right sides of the curve are mirror images.
• Mean, median, and mode are all equal and located at the center of the distribution.
• The curve tails off equally on both sides, extending infinitely.

In practical terms, when a sample follows a normal distribution, you can predict the behavior of the data within certain bounds. That is important for making inferences about a population from a sample. This is why, in Mattel Corporation's case, knowing that battery life follows a normal distribution aids in predicting how sample averages will behave.

Standard Error

The standard error (SE) is an important statistical measure used to understand how much sample means will vary from one another if multiple samples are taken. It is essentially the standard deviation of the sampling distribution of the mean.
You calculate the SE using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} \]where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
This measure helps in assessing the precision of the sample mean as an estimate of the population mean. A smaller standard error indicates that the sample mean is a more accurate reflection of the population mean. For the Mattel battery example, an SE of 1.1 hours means that, while individual battery lives can vary widely, the average lifespan across samples will have a much narrower spread.

Probability Distribution

Understanding a probability distribution is crucial for predicting how possible values of a random variable will occur. In essence, it defines how probabilities are distributed over the values of the random variable.
When dealing with continuous data, like battery life, the normal distribution is one example of a probability distribution. It allows us to find probabilities related to different outcomes by using the area under the curve.

• Probability distributions help in calculating how likely a particular sample mean will occur, using properties of the data itself.
• From the distribution curve, you determine the probability of a sample falling within a specific range by the area under the curve for that range.

This statistical tool is crucial for Sony’s sample testing program to assess the battery life's reliability in various ranges, such as between 34.5 and 36 hours.

Z-score

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It represents how many standard deviations an element is from the mean.
To compute the Z-score, use:\[ Z = \frac{X - \mu}{SE} \]where \( X \) is the observed value, \( \mu \) is the mean, and \( SE \) is the standard error.
In practical applications, the Z-score enables us to determine the probability of a score occurring within a normal distribution and to compare two scores from different distributions. For example, in the battery lifespan test, a calculated Z-score helps us understand whether a sample mean is typical or unusual compared to the overall distribution. A high Z-score means the sample mean is far from the population mean, while a low Z-score indicates it is closer. This helps in determining the likelihood of such an outcome and making predictions based on those results.