Problem 22 - Basic Statistics for Business and Economics

Question

The mean starting salary for college graduates in the spring of 2005 was \(\$ 36,280 .\) Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of \(\$ 3,300 .\) What percent of the graduates have starting salaries: a. Between \(\$ 35,000\) and \(\$ 40,000 ?\) b. More than \(\$ 45,000 ?\) c. Between \(\$ 40,000\) and \(\$ 45,000 ?\)

Step-by-Step Solution

Answer

(a) 52.25%, (b) 0.41%, (c) 12.51%

Step 1: Identify the Given Information

We are given that the mean starting salary is \( \mu = 36,280 \) dollars and the standard deviation is \( \sigma = 3,300 \) dollars. This suggests we need to use the normal distribution to find percentages related to salaries.

Step 2: Understanding Normal Distribution

Since salary distribution is normally distributed, we can use the standard normal distribution to find the required probabilities by converting given salaries to z-scores using the formula \( z = \frac{X - \mu}{\sigma} \).

Step 3: Calculate Z-scores for Part (a)

For salaries between \\(35,000 and \\)40,000, calculate z-scores. \[ z_{1} = \frac{35,000 - 36,280}{3,300} = -0.39 \] \[ z_{2} = \frac{40,000 - 36,280}{3,300} = 1.13 \]

Step 4: Find Probability for Part (a)

Using a standard normal distribution table or calculator, find the probabilities: - Probability(\(z<1.13\)) \( \approx 0.8708 \)- Probability(\(z<-0.39\)) \( \approx 0.3483 \)The probability between \\(35,000 and \\)40,000 is \(0.8708 - 0.3483 = 0.5225\) or 52.25%.

Step 5: Calculate Z-score for Part (b)

For salaries more than \$45,000, calculate the z-score.\[ z = \frac{45,000 - 36,280}{3,300} = 2.64 \]

Step 6: Find Probability for Part (b)

Using the standard normal distribution table or calculator, find the probability: - Probability(\(z>2.64\)) is \(1 - 0.9959 = 0.0041\) or 0.41%.

Step 7: Calculate Z-scores for Part (c)

For salaries between \\(40,000 and \\)45,000, calculate z-scores.\[ z_{1} = \frac{40,000 - 36,280}{3,300} = 1.13 \] \[ z_{2} = \frac{45,000 - 36,280}{3,300} = 2.64 \]

Step 8: Find Probability for Part (c)

Using a standard normal distribution table or calculator, find the probabilities: - Probability(\(z<2.64\)) \( \approx 0.9959 \)- Probability(\(z<1.13\)) \( \approx 0.8708 \)The probability between \\(40,000 and \\)45,000 is \(0.9959 - 0.8708 = 0.1251\) or 12.51%.

Key Concepts

Z-score

Imagine that a Z-score is a translation tool, allowing us to turn actual data points within our dataset into valuable positional information.

• The Z-score tells us how many standard deviations a particular data value is from the mean of the dataset.
• If a Z-score is 0, the data point is at the mean.
• A positive Z-score indicates that the data point is above the mean, while a negative Z-score means it's below.

To compute a Z-score, use the formula \( z = \frac{X - \mu}{\sigma} \), where \( X \) is your data point, \( \mu \) represents the mean, and \( \sigma \), the standard deviation. This gives a standardized way to evaluate how unusual or typical a particular income is, relative to the overall salary distribution.

Standard Deviation

Standard deviation is a crucial statistic that measures the amount of variation or dispersion in a dataset.

• It tells us how much individual data points tend to deviate from the mean.
• A small standard deviation implies that data points are generally close to the mean, creating a narrow distribution.
• Conversely, a large standard deviation signifies more spread-out data points.

In the context of our exercise, the standard deviation of \( \\(3,300 \) indicates how typical salaries deviate from the average salary of \( \\)36,280 \).Visualizing this can help understand the variations in the salary distribution and how different or similar they are to the average.

Probability Calculation

Calculating probability in a normal distribution allows us to answer questions like: How likely is a salary to fall within a certain range?

• Once Z-scores are determined, we can use them to find probabilities through a Z-table or a calculator, which shows the cumulative probability associated with each Z-score.
• For example, to find the probability of a salary between two values, subtract the smaller Z-score's probability from the larger one.
• To find the probability of salaries more than a certain amount, subtract the Z-score's cumulative probability from 1.

Applying this in our exercise, these calculations give insight into how salaries are distributed around the mean, showing how likely someone is to earn within specific salary brackets.