Problem 13 - Basic Statistics for Business and Economics

Question

A normal population has a mean of 20.0 and a standard deviation of \(4.0 .\) a. Compute the \(z\) value associated with 25.0 . b. What proportion of the population is between 20.0 and \(25.0 ?\) c. What proportion of the population is less than \(18.0 ?\)

Step-by-Step Solution

Answer

a. The Z-score for 25.0 is 1.25. b. About 39.44% of the population is between 20.0 and 25.0. c. About 30.85% of the population is less than 18.0.

Step 1: Understanding the Z-score Formula

To find the Z-score of a given value in a normal distribution, we use the formula: \[ Z = \frac{X - \mu}{\sigma} \] where \(X\) is the value for which we are calculating the Z-score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 2: Calculate the Z-score for 25.0

Given that \(X = 25.0\), \(\mu = 20.0\), and \(\sigma = 4.0\), we plug these into the Z-score formula: \[ Z = \frac{25.0 - 20.0}{4.0} = \frac{5.0}{4.0} = 1.25 \] So, the Z-score associated with 25.0 is 1.25.

Step 3: Interpret the Z-score for Proportion Calculation

Using the Z-score of 1.25, we refer to standard normal distribution tables (Z-tables) to find the proportion of the population below this Z-score, which is approximately 0.8944. This represents the proportion of the data less than 25.0.

Step 4: Find the Proportion between 20.0 and 25.0

The mean, 20.0, corresponds to a Z-score of 0, and from Z tables, about 0.5000 of the population is below 20.0. To find the proportion between 20.0 and 25.0, calculate \(0.8944 - 0.5000 = 0.3944\). Therefore, approximately 39.44% of the population is between 20.0 and 25.0.

Step 5: Calculate the Z-score for 18.0

For \(X = 18.0\), the Z-score is calculated as: \[ Z = \frac{18.0 - 20.0}{4.0} = \frac{-2.0}{4.0} = -0.5 \]

Step 6: Determine Proportion less than 18.0

Referencing the Z-table, a Z-score of -0.5 corresponds to a population proportion of approximately 0.3085. Thus, about 30.85% of the population is less than 18.0.

Key Concepts

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics. It represents a continuous probability distribution that is symmetric around its mean. The shape of the distribution resembles a bell, which is why it is often referred to as a "bell curve." In a normal distribution:

• The mean, median, and mode are all equal and lie at the center of the curve.
• The curve is symmetric about this mean, which means that it is evenly distributed on both sides.


• The spread of the curve is determined by the standard deviation, which measures the extent of variation or dispersion from the mean.
• Data near the mean are more frequent in occurrence, creating the bell shape.

Normal distribution is widely used in statistics because many variables in the natural world fit this pattern, especially when many small factors independently affect the values observed.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or spread in a set of data values. In the context of a normal distribution, it plays a crucial role in determining the shape and spread of the curve. Here are key aspects of standard deviation:

• A low standard deviation indicates that data points are close to the mean, which results in a sharp, peaked curve.
• A high standard deviation implies that data points are spread out over a wider range, which makes the bell curve flatter and wider.


• Standard deviation is denoted by the symbol \(\sigma\) (sigma).

When data follows a normal distribution, about 68% of all values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This is known as the "68-95-99.7 rule" or the empirical rule.

Proportion Calculation

Proportion calculation in a normal distribution helps us understand the percentage of data that lies within a certain range. This is essential when interpreting Z-scores, which measure how many standard deviations a value is from the mean. Let's see how this works:

• The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \), where \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.


• Once we calculate the Z-score, we use Z-tables (standard normal distribution tables) to find the corresponding proportion of the dataset that falls below that Z-score.
• This allows us to determine, for example, what fraction of a population scores between specific values, like between 20.0 and 25.0 or below a certain threshold like 18.0.

By understanding proportion calculations, we can make meaningful conclusions about the data and its distribution. These insights are invaluable in fields such as psychology, finance, and natural sciences.