Suppose The Data Concerning First-Year Salaries Of Graduates

5 Suppose The Data Concerning First Year Salaries Of Graduates Follow

Suppose the data concerning first-year salaries of graduates follows a normal distribution with the population mean μ = $60,000 and the standard deviation σ = $10,000. Denote X as the salary of a randomly selected graduate. (a) What is the probability that X is greater than $40,000? (b) What is the probability that X is between $30,000 and $90,000? (c) Suppose we interviewed 5 randomly selected SBU graduates and their salaries are X1, X2, ..., X5. Denote the average salaries of these 5 people as X̄ = (X1 + X2 + X3 + X4 + X5)/5. What is the expectation and the variance of X̄? (Hint: Use the standard normal table to answer (a) and (b).)

Paper For Above instruction

The analysis of first-year salaries of graduates often involves understanding the distribution and probability associated with these salaries. Assuming the data follows a normal distribution provides a framework for calculating various probabilities and statistical measures. In this paper, we consider the given population parameters and address the specific questions regarding probabilities and sample statistics.

Introduction

The salaries of recent graduates are influenced by multiple factors, including industry, education level, economic conditions, and geographic location. When aggregated across a large population, their distribution often approximates a normal distribution due to the Central Limit Theorem and the natural variation in earnings. Understanding the probabilities associated with different salary levels assists policymakers, educational institutions, and students in making informed decisions about career planning and economic forecasts.

Population Parameters and Assumptions

Given that the population mean (μ) is $60,000 with a standard deviation (σ) of $10,000, we assume that the salaries X of individual graduates are normally distributed: X ~ N(μ = 60,000, σ = 10,000). These parameters allow us to determine probabilities for specific salary ranges and to analyze the expected values and variances for sample means.

Part A: Probability that X is greater than $40,000

To compute P(X > $40,000), we first convert the raw salary to a standard normal variable (Z). The Z-score formula is:

Z = (X - μ) / σ

Plugging in the values for $40,000:

Z = (40,000 - 60,000) / 10,000 = -2

Using the standard normal table, P(Z -2) is:

P(X > 40,000) = 1 - P(Z

Therefore, there is approximately a 97.72% chance that a randomly selected graduate earns more than $40,000.

Part B: Probability that X is between $30,000 and $90,000

Next, for the salary range between $30,000 and $90,000, we compute the corresponding Z-scores:

Z₁ = (30,000 - 60,000) / 10,000 = -3

Z₂ = (90,000 - 60,000) / 10,000 = 3

From the standard normal table, P(Z

P(30,000

This indicates a 99.74% probability that a graduate's salary lies within this range.

Part C: Expected value and variance of the sample mean salary

Now, consider a sample of n=5 graduates, with individual salaries X_i for i=1 to 5, and their average salary X̄ = (X₁ + X₂ + X₃ + X₄ + X₅) / 5.

The expectation of the sample mean, E[X̄], is equal to the population mean:

E[X̄] = μ = $60,000

The variance of the sample mean, Var(X̄), is the population variance divided by the sample size n:

Var(X̄) = σ² / n = (10,000)² / 5 = 100,000,000 / 5 = 20,000,000

The standard deviation of the sample mean is thus:

SD(X̄) = √20,000,000 ≈ $4,472.14

These calculations demonstrate that, although individual salaries vary widely, the average salary of a small sample tends to be more precise and less variable, providing a more stable estimate of the true average.

Conclusion

Understanding the distribution of graduate salaries allows for effective probability estimations. The high probability that salaries exceed $40,000 underscores the economic viability for recent graduates. Additionally, the narrow confidence interval implied by the sample mean's variance emphasizes the reliability of average salary estimates in small samples. These statistical insights are valuable for stakeholders in education, employment sectors, and policymakers.

References

• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
• Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers and Scientists. Pearson.
• Moore, D. S., Notz, W., & Fligner, M. (2013). The Basic Practice of Statistics. W. H. Freeman.
• Hogg, R. V., Tanis, E. A., & Zimmerman, D. (2013). Probability and Statistical Inference. Pearson.
• Freund, J. E., & Williams, F. E. (2010). Modern Elementary Statistics. Pearson.
• Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.
• Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
• Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.