Problem 9 - Basic Statistics for Business and Economics

Question

The mean of a normal probability distribution is \(500 ;\) the standard deviation is \(10 .\) a. About 68 percent of the observations lie between what two values? b. About 95 percent of the observations lie between what two values? c. Practically all of the observations lie between what two values?

Step-by-Step Solution

Answer

a) 490 to 510 for 68%. b) 480 to 520 for 95%. c) 470 to 530 for practically all.

Step 1: Understanding the Problem

We are given a normal distribution with a mean, \( \mu = 500 \), and a standard deviation, \( \sigma = 10 \). We need to find the range of values for which different percentages of observations lie, based on the empirical rule.

Step 2: Applying the Empirical Rule for 68%

The empirical rule states that about 68% of observations fall within one standard deviation from the mean. Therefore, we calculate the range as follows: \( \mu - \sigma \) to \( \mu + \sigma \), which gives us \( 500 - 10 = 490 \) and \( 500 + 10 = 510 \). So the values range from 490 to 510.

Step 3: Applying the Empirical Rule for 95%

According to the empirical rule, about 95% of observations fall within two standard deviations from the mean. We compute this range as \( \mu - 2\sigma \) to \( \mu + 2\sigma \), resulting in \( 500 - 20 = 480 \) and \( 500 + 20 = 520 \). Hence, the range is from 480 to 520.

Step 4: Applying the Empirical Rule for Practically All Observations

Practically all observations, or about 99.7%, fall within three standard deviations from the mean, per the empirical rule. We find this range by computing \( \mu - 3\sigma \) to \( \mu + 3\sigma \), which yields \( 500 - 30 = 470 \) and \( 500 + 30 = 530 \). This indicates the range is from 470 to 530.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetrical, meaning it has a perfect bell-shaped curve when graphed. This type of distribution often arises in real-world phenomena because many factors contribute small amounts to measurements.
The key characteristic of a normal distribution is its symmetry around the mean. This means the left half is a mirror image of the right half. If you fold the distribution at the mean, both sides would perfectly overlap.
Normal distributions are fully defined by just two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)). These parameters describe the center and the spread of the distribution, respectively. When visualized, most of the data points in a normal distribution cluster around the mean.

Mean and Standard Deviation

The mean, often referred to as the average, is a measure of central tendency indicating where the center of the dataset lies. It is computed by adding up all data points and dividing by the number of points. In the context of a normal distribution, the mean provides the location of the center of the curve.
The standard deviation measures the amount of variation or dispersion in a set of values. In simple terms, it tells us how much the individual data points deviate from the mean. A smaller standard deviation indicates that data points are closer to the mean; a larger one shows more spread.
Together, the mean (\( \mu = 500 \)) and standard deviation (\( \sigma = 10 \)) succinctly describe our specific normal distribution. Adjusting these values changes the shape and position of the normal curve used to represent a dataset.

Percentage of Observations

In a normal distribution, the empirical rule provides an easy way to understand how data is spread relative to the mean. This rule states that for a normal distribution:

• About 68% of data falls within one standard deviation (\( \mu \pm \sigma \)) of the mean.
• Roughly 95% of data is within two standard deviations (\( \mu \pm 2\sigma \)) of the mean.
• Approximately 99.7% falls within three standard deviations (\( \mu \pm 3\sigma \)) of the mean.

Applying this to our example with a mean of 500 and a standard deviation of 10 gives the following ranges:

• 68% of observations are between 490 and 510.
• 95% are between 480 and 520.
• 99.7%, or practically all, are between 470 and 530.

These percentages help quickly assess the likelihood of an observation falling within a certain range, which is particularly useful for making predictions and understanding data behavior.