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Will pay right after the answer is posted. No Down Payment! Show work (i.e. U x-bax = xxx) Luz and Amber , of “He Wasn’t Stat Enough For Me! †found that the stress levels of statistics students before an exam is found to be normally distributed with a mean of 7.08 and a standard deviation of 0.63. Find the probability that the average stress level (on a scale 1 to 10) of 25 randomly selected “stats†students will be more than 6.25. What is the population mean? (1pts.) b.Solve the problem, using a bell curve. (3pts.)
Paper For Above instruction
Introduction
The statistical analysis of student stress levels provides valuable insights into the mental preparedness and anxiety associated with academic examinations. Understanding the distribution of stress levels among students can help educators design interventions to mitigate anxiety and improve performance. This paper addresses a specific problem involving the normal distribution of stress levels among statistics students, focusing on calculating the probability that a sample mean exceeds a certain threshold and conducting this analysis using a bell curve approach. The details of the problem are drawn from a scenario where the population parameters are known, and a sample is randomly selected for analysis.
Understanding the Problem and Population Parameters
According to Luz and Amber’s study expressed in "He Wasn’t Stat Enough For Me!", the stress levels of statistics students prior to exams follow a normal distribution characterized by a population mean (μ) and standard deviation (σ). Specifically, the population mean stress level is given as μ = 7.08, and the population standard deviation as σ = 0.63. These parameters describe the overall distribution of stress levels across the student population under consideration.
The problem asks for the probability that the average stress level for a sample of 25 students exceeds 6.25. It is essential to note that since the population standard deviation and mean are known, and the distribution is normal, the problem can be approached with classical statistical inference techniques, particularly using the properties of sampling distributions and the standard normal (z) distribution.
Methodology: Using the Bell Curve (Normal Distribution)
To solve this problem, the approach involves:
1. Identifying the sampling distribution of the sample mean.
2. Calculating the z-score corresponding to the sample mean of 6.25.
3. Using the standard normal distribution to find the probability that the sample mean exceeds 6.25.
The sampling distribution of the sample mean \(\bar{X}\) has a mean equal to the population mean \(\mu = 7.08\) and a standard error (SE) calculated by:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{0.63}{\sqrt{25}} = \frac{0.63}{5} = 0.126
\]
This standard error measures the variability of the sample mean from the population mean for samples of size 25.
Next, the z-score for the observed sample mean of 6.25 is computed as:
\[
z = \frac{\bar{x} - \mu}{SE} = \frac{6.25 - 7.08}{0.126} \approx \frac{-0.83}{0.126} \approx -6.59
\]
Using the standard normal distribution table or calculator, the probability associated with a z-score of -6.59 is effectively zero (p ≈ 0). Since the problem asks for the probability that the average stress level is more than 6.25, this is equivalent to:
\[
P(\bar{X} > 6.25) = 1 - P(Z
\]
Thus, there is an almost certain probability that the average stress level of the 25 students exceeds 6.25, given that the likelihood of a z-score of -6.59 is negligible.
Conclusion
The analysis demonstrates that the probability of the average stress level exceeding 6.25 among a sample of 25 statistics students is nearly certain, based on the known population parameters and the properties of the normal distribution. This example illustrates the power of the bell curve and normal distribution in statistical inference, especially when dealing with large samples and known population variances.
The calculation underscores that extreme deviations from the mean, such as a z-score of -6.59, are practically impossible under normal distribution assumptions. Hence, the probability that the sample mean is greater than 6.25 approaches one, reinforcing the reliability of using the normal distribution for such inferential statistics.
References
- Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics (9th ed.). W.H. Freeman.
- Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data (4th ed.). Pearson.
- Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical Statistics with Applications (7th ed.). Cengage Learning.
- Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
- Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
- Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
- Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
- Lind, D. A., Marchal, W. G., & Wathen, S. A. (2014). Statistical Techniques in Business & Economics (16th ed.). McGraw-Hill Education.
- Bluman, A. G. (2017). Elementary Statistics: A Step By Step Approach (9th ed.). McGraw-Hill Education.