Problem 16 - Basic Statistics for Business and Economics

Question

The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. a. What is the area between 415 pounds and the mean of 400 pounds? b. What is the area between the mean and 395 pounds? c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

Step-by-Step Solution

Answer

a) 0.4332, b) 0.1915, c) 0.3085

Step 1: Understanding the Question and Given Information

Here, we are dealing with a normal distribution where the mean (\(\mu\)) is 400 pounds and the standard deviation (\(\sigma\)) is 10 pounds. We are given three parts to solve, each involves determining probabilities related to the normal distribution.

Step 2: Calculating the Z-Score for 415 Pounds

To find the area between 415 pounds and the mean, we first calculate the Z-score for 415. The Z-score formula is \(Z = \frac{X - \mu}{\sigma}\). Substituting 415 for \(X\), \(400\) for \(\mu\), and \(10\) for \(\sigma\), we get \(Z = \frac{415 - 400}{10} = 1.5\).

Step 3: Finding the Area Between the Mean and 415 Pounds

For a Z-score of 1.5, use a Z-table to find the area between 0 and 1.5. The area associated with Z = 1.5 is approximately 0.4332. This area represents the probability or area under the normal curve between the mean and 415 pounds.

Step 4: Calculating the Z-Score for 395 Pounds

Next, calculate the Z-score for 395 using the formula \(Z = \frac{X - \mu}{\sigma}\). With \(X = 395\), \(\mu = 400\), and \(\sigma = 10\), we find \(Z = \frac{395 - 400}{10} = -0.5\).

Step 5: Finding the Area Between the Mean and 395 Pounds

For Z = -0.5, use a Z-table, which provides the area from the mean to Z = -0.5. This area is approximately 0.1915, representing the probability that a value is between 400 and 395 pounds.

Step 6: Finding Probability of Less Than 395 Pounds

From a Z-table, a Z-score of -0.5 gives the cumulative area from the left of the mean to -0.5, which is 0.3085. This value represents the probability of selecting a value less than 395 pounds.

Key Concepts

Z-score

The Z-score is a crucial concept in statistics, especially in the context of the normal distribution. It tells us how far away a particular value is from the mean, measured in standard deviations. To calculate the Z-score, you use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is the data point in question, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation. This transformation standardizes our data point, allowing us to use the standard normal distribution table (Z-table) to find probabilities and areas under the curve.

• A positive Z-score indicates the value is above the mean.
• A negative Z-score means it is below the mean.
• A Z-score of zero means the value is exactly at the mean.

In our example, for a weight of 415 pounds, we calculated a Z-score of 1.5, meaning it is 1.5 standard deviations above the mean of 400 pounds. For 395 pounds, the Z-score is -0.5, indicating it is 0.5 standard deviations below the mean.

Probability

Probability is the measure of how likely an event is to occur. In the context of the normal distribution, it represents the area under the curve within a specific range. Understanding probabilities with the normal distribution involves calculating the Z-score and then using a Z-table. The Z-table gives you the probability that a value falls between the mean and a specified point on the curve.

• The area between two points represents the probability of a value falling within that range.
• To find the probability of a value being less than a point, find the cumulative area to the left of that Z-score.

In the given exercise:

• The probability of picking a value between 400 and 415 pounds is the area under the curve from the mean to Z = 1.5, which is approximately 0.4332.
• The probability of a value falling between the mean and 395 pounds is about 0.1915.
• The likelihood of selecting a random value less than 395 pounds is about 0.3085.

These E-values, found from the areas in the Z-table, help you understand the relative likelihoods of occurrences within a distribution.

Standard Deviation

Standard deviation is a fundamental measure in statistics that quantifies the amount of variation or dispersion in a set of values. In a normal distribution, it helps determine the width or "spread" of the curve. The formula for standard deviation is \[ \sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \] where \(X_i\) are the individual data points, \(\mu\) is the mean, and \(N\) is the number of data points.

• A smaller standard deviation indicates that data points tend to be close to the mean.
• A larger standard deviation means the data is spread out over a wider range of values.

In the context of the problem, the standard deviation of 10 pounds tells us how much the weights vary from the average of 400 pounds. Knowing this, along with the Z-score, we can easily calculate the probability of different scenarios such as the likelihood of weights falling below certain values or between specific ranges.