Homework 4

Probability Distributions

Important

Due Friday, Oct 3, 11:59 PM

• Homework 4 covers Week 5 and 6.

• Please submit your work in one PDF file to D2L > Assessments > Dropbox. Multiple files or a file that is not in pdf format is not allowed.

• Handwritten tables and figures receive no credits.

• You do not need to attach your code of any language you use in the homework. However, if you fail to complete your calculations or produce your table or figure, you receive partial credits if your code is attached.

• Questions started with (MSSC) are required for MSSC 5720 students, and optional for MATH 4720 students.

Homework Questions

1. What percent of a standard normal distribution \(N(\mu=0, \sigma=1)\) is found in each region? Draw a normal graph may help.
   1. \(-0.4 < Z < 1.5\)
   2. \(|Z| > 2\)

2. The average daily high temperature in June in Milwaukee is 77\(^{\circ}\) F with a standard deviation of 5\(^{\circ}\) F. Suppose that the temperatures in June closely follow a normal distribution.
   1. What is the probability of observing an 83\(^{\circ}\) F temperature or higher in Milwaukee during a randomly chosen day in June?
   2. How cool are the coldest 10% of the days (days with lowest average high temperature) during June in Milwaukee?

3. Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 70% of 18-20 year olds consumed alcoholic beverages in any given year.
   1. Suppose a random sample of ten 18-20 year olds is taken. When does it make sense to use binomial distribution for calculating the probability that exactly six consumed alcoholic beverages?
   2. What is the probability that exactly four out of ten 18-20 year olds have consumed an alcoholic beverage?
   3. What is the probability that at most 2 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?

4. A Starbucks in Chicago serves an average of 75 customers per hour during the morning rush.
   1. Which distribution have we studied that is most appropriate for calculating the probability of a given number of customers arriving within one hour during this time of day?
   2. Calculate the probability that this Starbucks serves 70 customers in one hour during this time of day.

5. (MSSC) In observing patients administered a new drug product in a properly conducted clinical trial, the number of persons experiencing a particular side effect might be quite small. Suppose \(\pi\) (the probability a person experiences a side effect to the drug) is \(.001\) and \(1,000\) patients in the clinical trial received the drug.
   1. Use binomial distribution to Compute the probability that none of a random sample of \(n = 1,000\) patients administered the drug experiences a particular side effect.
   2. Use Poisson distribution to Compute the probability that none of a random sample of \(n = 1,000\) patients administered the drug experiences a particular side effect. [Hint:] The parameter \(\lambda = n\pi\).
   3. When \(n\) is large and \(\pi\) is small, Poisson\((\lambda)\) looks pretty similar to binomial\((n, \pi)\). Plot the two distributions for \(x = 0, 1, \dots, 8\). Are they similar?

6. AI Usage Declaration. Using GenAI is permitted for this course. If you choose to use GenAI to assist with your homework, you must include a brief statement documenting your use. Please provide the following information:
   1. Why/How I Used AI Why do you need to use GenAI? Which tool did you use? Describe your prompts or questions. What and how did you ask the AI to help you?
   2. Generated Output Include a screenshot or excerpt (copy and paste) of the AI’s response.
   3. How I Used the Output Did you revise it? Did you use it directly, or compare it with your answers? What decisions did you make based on the output?