Problem 15 - Basic Statistics for Business and Economics

Question

A recent study of the hourly wages of maintenance crew members for major airlines showed that the mean hourly salary was \(\$ 20.50,\) with a standard deviation of \(\$ 3.50 .\) If we select a crew member at random, what is the probability the crew member earns: a. Between \(\$ 20.50\) and \(\$ 24.00\) per hour? b. More than \(\$ 24.00\) per hour? c. Less than \(\$ 19.00\) per hour?

Step-by-Step Solution

Answer

a. 0.3413 b. 0.1587 c. 0.3336

Step 1: Understand the Normal Distribution

Since the hourly wages of a maintenance crew member are normally distributed, we can use the properties of the normal distribution to find probabilities. The mean is given as \( \mu = 20.50 \) and the standard deviation as \( \sigma = 3.50 \).

Step 2: Calculate Z-scores

For any salary \( x \), the Z-score is calculated using the formula: \[ Z = \frac{x - \mu}{\sigma} \]- For \( x = 24.00 \): \( Z = \frac{24.00 - 20.50}{3.50} = 1.00 \)- For \( x = 20.50 \): \( Z = \frac{20.50 - 20.50}{3.50} = 0.00 \)- For \( x = 19.00 \): \( Z = \frac{19.00 - 20.50}{3.50} = -0.43 \)

Step 3: Use Z-table for Probability Calculations

- Probability for between \\(20.50 and \\)24.00: \[ P(0.00 < Z < 1.00) = P(Z < 1.00) - P(Z < 0.00) \] From Z-tables: \( P(Z < 1.00) = 0.8413 \) and \( P(Z < 0.00) = 0.5000 \). So, the probability is \( 0.8413 - 0.5000 = 0.3413 \).- Probability for more than \\(24.00: \[ P(Z > 1.00) = 1 - P(Z < 1.00) \] \( P(Z > 1.00) = 1 - 0.8413 = 0.1587 \).- Probability for less than \\)19.00: \[ P(Z < -0.43) \] From Z-tables: \( P(Z < -0.43) = 0.3336 \).

Key Concepts

Z-score

The Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. In simpler terms, it tells you how many standard deviations the original value is away from the average. To calculate the Z-score, you can use the formula: \[ Z = \frac{x - \mu}{\sigma} \]

• \( x \) is the value you are analyzing.
• \( \mu \) is the mean of the data.
• \( \sigma \) is the standard deviation.

For example, if a maintenance crew member earns \\(24.00 (\( x = 24.00 \)), and the mean salary is \\)20.50 with a standard deviation of \$3.50, we calculate:\[ Z = \frac{24.00 - 20.50}{3.50} = 1.00 \]This Z-score of 1.00 means that the salary is exactly 1 standard deviation above the mean. Z-scores are crucial for understanding where a value stands in a normal distribution.

Probability Calculations

When dealing with normal distribution, probability calculations help us understand how likely it is for a random value to fall within a certain range or beyond a specific point.

Steps for Calculating Probability

1. **Calculate the Z-score**: This is a necessary step as it converts the variable into a standard normal variable. 2. **Use the Z-table**: The Z-table provides the probability that a standard normal variable is less than the given Z-score. For instance, from the Z-table: - Probability for \( Z < 1.00 \) is 0.8413. - Probability for \( Z < -0.43 \) is 0.3336.

Interpreting the Results

- **Between Two Values**: To find the probability of earning between \\(20.50 and \\)24.00, you subtract the smaller Z-score probability from the larger one: \[ P(0.00 < Z < 1.00) = 0.8413 - 0.5000 = 0.3413 \] - **Above a Value**: For a salary more than \\(24.00, calculate: \[ P(Z > 1.00) = 1 - 0.8413 = 0.1587 \]- **Below a Value**: For less than \\)19.00, you use the Z-score directly from the Z-table: \[ P(Z < -0.43) = 0.3336 \] Probability calculations using the Z-score and the Z-table transform real-world data into insights, helping us predict outcomes based on the data's normal distribution.

Standard Deviation

Standard deviation is a measure that indicates how much variation or dispersion there is from the mean in a set of data. A smaller standard deviation means that the data points are generally close to the mean, while a larger one indicates more spread out data.In our exercise, the standard deviation is \\(3.50, which is used to measure how salaries of crew members deviate from the average salary of \\)20.50. Here’s why it’s important:

• It provides a scale to quantify the volatility or risk in a dataset.
• By knowing the standard deviation, we can understand what to expect when randomly picking a data point.

The combination of using standard deviation with other concepts like the Z-score allows for assessing probabilities and understanding the normal distribution better. Why does this matter? In practical terms, if you're looking at crew salaries, knowing the standard deviation helps identify what range of salaries is typical and which salaries are outliers. This insight can guide policy-making and salary adjustment decisions.