Problem 51 - Basic Statistics for Business and Economics

Question

The annual sales of romance novels follow the normal distribution. However, the mean and the standard deviation are unknown. Forty percent of the time sales are more than 470,000 , and 10 percent of the time sales are more than \(500,000 .\) What are the mean and the standard deviation?

Step-by-Step Solution

Answer

The mean is approximately 462,610 and the standard deviation is approximately 29,172.

Step 1: Understand the Problem

The problem provides two key pieces of information about sales distribution: 40% of sales are above 470,000 and 10% of sales are above 500,000. We are to find the mean and standard deviation of the sales distribution.

Step 2: Convert to Z-Scores

Start by finding the Z-scores corresponding to the proportions given. For a normal distribution, we know: - 40% above corresponds to a Z-score of about 0.2533 (consult standard normal distribution tables or Z-score calculators). - 10% above corresponds to a Z-score of about 1.2816. These values tell us how many standard deviations above the mean these points lie.

Step 3: Set Up the Equations

Using the Z-score formula, \[ Z = \frac{X - \mu}{\sigma} \]we get two equations:1. \(0.2533 = \frac{470,000 - \mu}{\sigma}\)2. \(1.2816 = \frac{500,000 - \mu}{\sigma}\)

Step 4: Solve the Equations

Rearrange and solve the equations:From equation 1, \(470,000 - \mu = 0.2533\sigma\).From equation 2, \(500,000 - \mu = 1.2816\sigma\).Subtract the first equation from the second:\[(500,000 - 470,000) = (1.2816 - 0.2533)\sigma\]\[30,000 = 1.0283\sigma\]So, \[\sigma = \frac{30,000}{1.0283} \approx 29,171.93\]

Step 5: Calculate the Mean

Substitute the value of \(\sigma\) back into one of the original equations (for simplicity, use the first):\[470,000 - \mu = 0.2533 \times 29171.93\]\[470,000 - \mu \approx 7390.07 \]\[\mu = 470,000 - 7390.07 \]\[\mu \approx 462,609.93\]

Step 6: Verify the Solution

Double-check calculations and ensure the numbers make logical sense given the problem's context. We calculated \(\mu \approx 462,610\) and \(\sigma \approx 29,172\), which align with a normal distribution's expected range.

Key Concepts

Z-scores

Z-scores are essential in statistics for understanding how far away a particular data point is from the mean in a standard normal distribution.

• A Z-score signals the number of standard deviations a point lies from the mean.
• If the Z-score is positive, the data point is above the mean; if negative, it is below it.
• Understanding Z-scores helps compare data points across different normal distributions.

To convert a data point (X) into a Z-score, we use the formula: \( Z = \frac{X - \mu}{\sigma} \)
Here, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.
In our exercise, we found that 40% of sales exceeding 470,000 corresponds to a Z-score of 0.2533, while 10% exceeding 500,000 corresponds to a Z-score of 1.2816. These Z-scores help in setting up equations to find unknown parameters like the mean and standard deviation.

mean calculation

The mean, represented as \( \mu \), is a measure of the central tendency of a normal distribution.

• The mean provides us with a single value that summarizes the entire dataset.
• For symmetric distributions like normal distributions, the mean is also the point of symmetry.
• It can be thought of as the balancing point of the distribution.

In the provided exercise, the mean of sales is initially unknown. However, with the help of Z-scores and sales values, we can calculate it.
The equation we set up using the Z-score for 470,000 sales \( (0.2533) \) is:
\[470,000 - \mu = 0.2533 \times \sigma\]
Using the known standard deviation value (calculated as approximately 29,171.93), we substitute back into the equation to find the mean:\[\mu = 470,000 - 0.2533 \times 29,171.93 \approx 462,609.93\]
Thus, the calculated mean sales is about 462,610.

standard deviation calculation

The standard deviation \( \sigma \) quantifies how much variation or dispersion exists from the mean in a dataset.

• A larger standard deviation indicates more spread out data points, while a smaller one shows data points cluster closely around the mean.
• In a normal distribution, around 68% of data points lie within one standard deviation of the mean.
• About 95% lie within two standard deviations, highlighting predictability.

In the exercise, the standard deviation was initially unknown. From our Z-score equations:\[\sigma = \frac{30,000}{1.0283} \approx 29,171.93\]
This was derived by comparing the sales above specific thresholds and understanding that each threshold maps to a different Z-score.
Calculating the standard deviation provides insight into the reliability and spread of the sales figures, crucial for business strategies.