Name HW 6 Chapter 5 Due 2/27/51 We Would Like To Use

Name Hw 6 Chapter 5due 227151 We Would Like To Use

We would like to use Minitab to help simulate a large number of repetitions of the probability experiment of rolling a die. These observations will then be used to make conclusions about the probability of the outcomes of this probability experiment. In Minitab, use Calc>Random Data>Integer… to simulate 12,000 “rolls” of a six-sided die. Then, to help count the rolls, go to Graph>Bar Chart… to make a frequency chart. Hover your mouse over the bars in the chart to get the totals. Copy-and-paste your frequency graph here.

Outcome | Number of Rolls | Probability

Roll a ... Identify the “method” used in #1. How will your answers for the probabilities in #1 compare to the answers of other students in our class? Will they be exactly the same, very different, a little different, or something else?

Now, complete the table using the fact that each outcome for a roll is equally likely. Outcome | Probability | Roll a ... Identify the “method” used in #3 to compute the probabilities. How will your answers for the probabilities in #3 compare to the answers of other students in our class? Will they be exactly the same, very different, a little different, or something else?

According to The Law of Large Numbers, how should your answers to #1 and #3 compare?

Let E be the event that a randomly selected person in the U.S. who is over the age of 24 has attended college. Let F be the event that a randomly selected U.S. resident over the age of 24 is female. P(E) = 0.559, P(F) = 0.518, P(E and F) = 0.295.

• a. What is the probability that a randomly selected person in the U.S. over the age of 24 is a male?
• b. What is the probability that a randomly selected person in the U.S. over the age of 24 has not attended college?
• c. What is the probability that a randomly selected person in the U.S. over the age of 24 is a female or has attended college?

What part of the homework assignment did you find the most difficult to complete? Are there any parts of the homework that still give you difficulties?

Paper For Above instruction

The assignment involved both simulation and probability calculations centered around a die-rolling experiment, coupled with real-world application of probability concepts relating to demographic data. This comprehensive exercise aimed at understanding probability distributions through simulation, comparative analysis, and theoretical calculations grounded in the law of large numbers and basic probability rules.

Simulating the Die Roll Experiment Using Minitab

The first part of the homework required utilizing Minitab software to simulate 12,000 rolls of a six-sided die. Employing the function Calc>Random Data>Integer, I set parameters to generate integers from 1 to 6, representing the outcomes of each die roll. After the simulation, a bar chart was created via Graph>Bar Chart to visualize the frequency of each outcome.

The generated bar chart revealed the distribution of outcomes across the simulated rolls. Hovering over the bars indicated the count of each outcome, which provided empirical data for calculating probabilities. The outcome frequencies were divided by 12,000 to determine the experimental probabilities of each result.

This simulation is a practical application of probability theory, offering insights into the experimental frequency of each outcome against the theoretical probability, which is 1/6 (~0.1667) for each face of the die.

Analysis of Probabilities and Methods

In the second step, the method used to compute the probabilities from the simulation was based on relative frequency, calculated by dividing each outcome's count by the total number of rolls (12,000). This empirical approach approximates the true probability based on observed data. The expected theoretical probability for each face, given the die's fairness, is exactly 1/6. The comparison between empirical and theoretical probabilities is useful in understanding how close a large sample size's outcomes align with expected probabilities.

Given the law of large numbers, as the number of simulated rolls increases, the empirical probabilities should converge toward the theoretical probability of 1/6. Variations between the two are expected in smaller samples but diminish as the number of repetitions grows. In this context, the simulated data with 12,000 rolls is sufficiently large for the empirical probabilities to closely approximate the theoretical value, illustrating the law's principle.

Probability Calculations in Real-World Context

The latter part of the assignment involved applying probability principles to demographic data from the U.S. population over age 24. The provided probabilities included P(E) = 0.559 for attending college, P(F) = 0.518 for being female, and P(E and F) = 0.295 for being female and attending college. Using these, several questions were answered:

• a. The probability that a person is male is calculated as P(male) = 1 - P(F) = 1 - 0.518 = 0.482. This complements the given probability of being female, adhering to the complement rule in probability theory.
• b. The probability that a person has not attended college is P(not E) = 1 - P(E) = 1 - 0.559 = 0.441. This uses the complement rule directly.
• c. To find the probability that a person is female or attended college, the inclusion-exclusion principle is applied: P(F or E) = P(F) + P(E) - P(E and F) = 0.518 + 0.559 - 0.295 = 0.782.

This analysis underscores how basic probability rules like complements and the inclusion-exclusion principle can be applied to real-world demographic data, providing insight into population characteristics.

Reflection and Difficulties

One of the most challenging parts of this assignment was accurately simulating large data sets in Minitab and correctly interpreting the resulting bar chart. Ensuring the correct use of the software functions and understanding how to convert frequency counts into probabilities required careful attention. Additionally, applying probability rules to population data demanded clarity in understanding the concepts of joint, marginal, and union probabilities.

Some difficulties still persist in fully grasping the implications of the law of large numbers in smaller samples versus large samples, and how slight deviations in empirical data influence the understanding of theoretical probabilities. Continuous review of probability principles and further practice with statistical software can help overcome these challenges.

References

• Moore, D. S., Notz, W. I., & Fligner, M. A. (2019). The Basic Practice of Statistics (8th ed.). W.H. Freeman.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications (Vol. 1). Wiley.
• Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis (6th ed.). Pearson.
• Shao, J. (2003). Mathematical Statistics. Springer.
• Agresti, A. (2018). Statistical Thinking: Improving Business Performance (2nd ed.). CRC Press.
• Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press.
• U.S. Census Bureau. (2020). Population Statistics and Demographic Data. https://www.census.gov
• MTI Data & Resources. (2023). Minitab Statistical Software Documentation. https://support.minitab.com