Business Weekly Conducted A Survey Of Graduates From 30 Top
Business Weekly Conducted A Survey Of Graduates From 30 Top Mba Progra
Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is 147000 dollars. Assume the standard deviation is 35000 dollars. Suppose you take a simple random sample of 61 graduates. Find the probability that a single randomly selected graduate has a salary between 158203.2 and 164028.9 dollars. P (158203.2
Paper For Above instruction
The problem involves calculating probabilities related to the salary distribution of MBA graduates using normal distribution concepts. Specifically, the task involves two probability calculations: one for a single graduate's salary and another for the mean salary of a sample of graduates.
Understanding the Data and Parameters
The survey indicates that the mean annual salary among 10-year MBA graduates is $147,000, with a standard deviation of $35,000. These parameters suggest that salaries are approximately normally distributed, a common assumption in statistical analyses of income data due to the Central Limit Theorem and empirical observations of income distributions.
Calculating the Probability for a Single Graduate
First, we want to find the probability that a randomly selected graduate's salary (denoted by X) falls between $158,203.20 and $164,028.90. Since the salary distribution is assumed to be normal, we standardize these values to Z-scores using the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
where:
- \( \mu = 147,000 \),
- \( \sigma = 35,000 \).
Calculating the Z-scores:
\[
Z_1 = \frac{158,203.20 - 147,000}{35,000} \approx \frac{11,203.20}{35,000} \approx 0.3201
\]
\[
Z_2 = \frac{164,028.90 - 147,000}{35,000} \approx \frac{17,028.90}{35,000} \approx 0.4865
\]
Using standard normal distribution tables or software like R, Python, or a calculator, we find the probabilities corresponding to these Z-scores:
\[
P(Z
\]
\[
P(Z
\]
Therefore, the probability that a single graduate's salary is between $158,203.20 and $164,028.90 is:
\[
P(158203.2
\]
Answer for single salary probability: 0.0607
---
Calculating the Probability for the Sample Mean
Next, we consider the probability that the mean salary (\( M \)) of a sample of size \( n = 61 \) graduates falls within the same salary range. The sampling distribution of the sample mean is also normal with:
- Mean \( \mu_M = \mu = 147,000 \),
- Standard error \( SE = \frac{\sigma}{\sqrt{n}} = \frac{35,000}{\sqrt{61}} \).
Calculating the standard error:
\[
SE = \frac{35,000}{\sqrt{61}} \approx \frac{35,000}{7.8102} \approx 4486.49
\]
Standardizing the bounds for the sample mean:
\[
Z_1 = \frac{158,203.20 - 147,000}{4486.49} \approx \frac{11,203.20}{4486.49} \approx 2.499
\]
\[
Z_2 = \frac{164,028.90 - 147,000}{4486.49} \approx \frac{17,028.90}{4486.49} \approx 3.796
\]
Using normal distribution tables or software:
\[
P(Z
\]
\[
P(Z
\]
Thus, the probability that the sample mean falls between the specified bounds is:
\[
P(158203.2
\]
Answer for the sample mean probability: 0.0061
---
Conclusion
The probability that a single randomly selected graduate has a salary between $158,203.20 and $164,028.90 is approximately 0.0607. In contrast, the probability that the mean salary of a sample of 61 graduates falls within the same interval is much lower, approximately 0.0061, reflecting the Law of Large Numbers and the decreased variability of the sample mean.
References
- Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.
- Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
- Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
- Mooney, H. M., & Duval, R. D. (1993). Bootstraps: A guide to experimental and computational statistics. Springer.
- McClave, J. T., & Sincich, T. (2012). Statistics. Pearson Education.
- Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
- Ross, S. (2014). Introduction to Probability Models. Academic Press.
- Bickel, P. J., & Doksum, K. A. (2015). Mathematical Statistics: Basic Ideas and Selected Topics. CRC Press.
- Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.
- Agresti, A. (2018). Statistical Thinking: Improving Business Performance. CRC Press.