Problem 52 - An Introduction to Mathematical Statistics and Its Applications

Question

Records for the past several years show that the amount of money collected daily by a prominent televangelist is normally distributed with a mean \((\mu)\) of \(\$ 20,000\) and a standard deviation \((\sigma)\) of \(\$ 5000\). What are the chances that tomorrow's donations will exceed \(\$ 30,000 ?\)

Step-by-Step Solution

Answer

The probability that tomorrow's donations will exceed \$30,000 is \(2.28\%\).

Step 1: Understanding the problem

In this problem, you are given a daily collection that follows a normal distribution, with a mean of \$20,000 and a standard deviation of \$5,000. You are asked to find the probability of tomorrow's donation exceeding \$30,000.

Step 2: Calculate the Z-score

The first step is to calculate the Z-score. The formula to calculate the Z-score is \(Z = (X - µ)/σ\) where \(X\) = \$30,000 is the target value, \(µ\) = \$20,000 is the mean, and \(σ\) = \$5,000 is the standard deviation. Substituting the values gives us \(Z = (\$30,000 - \$20,000) / \$5,000 = 2.0\).

Step 3: Find the probability

Next, we utilize the standard normal distribution table to find the probability associated with the \(Z\)-score. For a \(Z\)-score of \(2.0\), the table gives us a probability of \(0.9772\) or \(97.72\%\). This is the proportion of data below the \(Z\)-score of \(2.0\). However, the problem asks for the probability of donations exceeding \$30,000 so the probability becomes \(1 - 0.9772 = 0.0228\) or \(2.28\%\).

Key Concepts

Normal Distribution

The concept of *Normal Distribution* plays an essential role in statistics and probability. It represents a continuous probability distribution that is symmetrical, forming the classic bell shape curve. In a normal distribution, the mean, median, and mode of the data set are all the same, lying exactly at the center of the distribution.

• The values tend to cluster around the mean.
• The spread of these values is described by the standard deviation.
• The tails of the distribution approach, but never touch, the horizontal axis.

In the problem you studied, donations are said to be normally distributed with a mean donation amount of $20,000. This means on average, the televangelist collects around this amount. The normal distribution helps us predict the likelihood of various outcomes, such as the probability of collecting more or less than a certain amount in donations within a given standard deviation. Understanding normal distribution enables you to establish how unusual or common certain data points are, relative to the average.

Z-score

The *Z-score* is a powerful tool in statistics that allows you to determine how far away a specific data point is from the mean, in terms of the number of standard deviations. This makes it easier to compare different data points from various normal distributions.
The formula to calculate the Z-score is: \[Z = \frac{(X - \mu)}{\sigma}\] Here:

• \(X\) is the raw score you are investigating (in this case, \(30,000).
• \(\mu\) is the mean of the normal distribution (\)20,000).
• \(\sigma\) is the standard deviation (\(5,000).

In the exercise provided, the Z-score was determined to be 2.0. This indicates the donation amount of \)30,000 is two standard deviations above the mean amount of $20,000. The Z-score helps us assess how usual or unusual a specific donation amount would be relative to the expected average donation.

Standard Deviation

*Standard Deviation* is a key concept in statistics, serving as a measure of the dispersion or variability in a data set. A low standard deviation suggests that the data points are close to the mean, while a high standard deviation indicates a wider spread. This concept provides insight into how dispersed the values of a data set are around the mean.

• It's calculated as the square root of the variance.
• In normally distributed data, about 68% of values fall within one standard deviation of the mean.
• Approximately 95% of values lie within two standard deviations.
• 99.7% are within three standard deviations.

In the context of the exercise, the standard deviation was $5,000. This indicates the typical gap between any day's donations and the average. Understanding standard deviation helps you comprehend the variability within the donations collected, highlighting whether the amount collected on any given day is close to or far from the average collection.