Econ 2113 Business Statistics Final Assignment For Students
Econ 2113 Business Statistics Final Assignment Students in A College Cou
Analyze the given exam scores that are normally distributed with a mean of 76 and a standard deviation of 8. Calculate the z-scores for Bob (score of 88) and Tom (score of 69). Determine the area under the normal curve between Bob's z-score and the mean, converting this area to a percentage. Find the percentage of students scoring lower than Bob and lower than Tom, showing all work.
Paper For Above instruction
The analysis of exam scores using the properties of the normal distribution enables educators and statisticians to understand performance benchmarks and student performance distributions effectively. This paper interprets given exam scores within the framework of standard normal distribution calculations to elucidate how individual performances relate to the overall student body.
Given that exam scores are normally distributed with a mean (μ) of 76 and a standard deviation (σ) of 8, it becomes essential to evaluate individual scores in terms of their z-scores. The z-score, defined as the number of standard deviations a data point is from the mean, provides a standardized way to compare individual scores across different distributions.
Part a: Calculating Bob's z-score
Bob scored 88. To determine his z-score, the formula is:
z = (X - μ) / σ
Substituting the values:
z = (88 - 76) / 8 = 12 / 8 = 1.5
Rounding to the nearest hundredth, Bob's z-score is 1.50. This indicates Bob's score is 1.5 standard deviations above the mean.
Part b: Calculating Tom's z-score
Tom scored 69. Applying the same formula:
z = (69 - 76) / 8 = (-7) / 8 = -0.875
Rounded to the nearest hundredth, Tom's z-score is -0.88. This situates Tom's score slightly below the population mean, nearly one standard deviation below.
Part c: Area between Bob's z-score and the mean
The mean corresponds to a z-score of 0. The area between Bob's z-score (1.50) and the mean (0) can be obtained from standard normal distribution tables or computational tools. The cumulative area to z = 1.50 is approximately 0.9332, meaning about 93.32% of scores fall below Bob's score. The area below the mean (z=0) is always 0.5 (50%).
Therefore, the area between the mean and Bob's z-score is:
0.9332 - 0.5 = 0.4332
This is approximately 43.32%. Converting to a percentage, about 43.32% of students scored between the mean and Bob's score.
Part d: Percentage of students scoring lower than Bob
From the cumulative distribution function (CDF), the area below Bob's z-score (1.50) is approximately 0.9332 or 93.32%. Hence, about 93.32% of students scored lower than Bob.
Part e: Percentage of students scoring lower than Tom
Similarly, the area below Tom's z-score (-0.88) from the standard normal table is approximately 0.1894. Hence, about 18.94% of students scored lower than Tom.
Conclusion
These calculations establish a clear understanding of relative student performance. Bob's high score places him well above the average, with a significant percentage of students scoring below him. Conversely, Tom's score places him below average, with a sizable portion of students performing better. Such analysis aids educators in identifying percentile ranks and preparing targeted interventions where necessary.
References
- Ott, L. (2012). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
- Devore, J. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
- Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications. Cengage Learning.
- Freeman, J. (2014). Univariate and Multivariate Statistical Process Control. CRC Press.
- NIST/SEMATECH. (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology.
- Bluman, A. G. (2018). Elementary Statistics: A Step By Step Approach. McGraw-Hill Education.
- Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Academic Press.
- Wilkinson, L. (2012). The Grammar of Graphics. Springer Science & Business Media.
- Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.