Homework 5

Inferences about population means

Important

Due Friday, Oct 24, 11:59 PM

• Homework 5 covers Week 7 and 8.

• 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.

• 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. Assume that females have pulse rates that are normally distributed with a mean of 74.0 beats per minute and a standard deviation of 12.5 beats per minute.
   1. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 80 beats per minute.
   2. If 16 adult female are randomly selected, find the probability that their mean pulse rate is less than 80 beats per minute.
   3. Why can the normal distribution be used in (b), even though the sample size does not exceed 30?

2. An independent random sample is selected from an approximately normal population with unknown standard deviation. Find the degrees of freedom and the critical \(t\)-value for the given sample size and confidence level.
   1. \(n = 6\), confidence level 90%
   2. \(n = 21\), confidence level 98%

3. Here are summary statistics for randomly selected weights of newborn girls: \(n =205\), \(\bar{x} = 30.4\)hg (1hg = 100 grams), \(s = 7.1\)hg.
   1. Compute a 95% confidence interval for \(\mu\), the mean weight of newborn girls.
   2. Are the result in (a) very different from the 95% confidence interval if \(\sigma = 7.1\)? Why?

4. A 95% confidence interval for a population mean \(\mu\) is given as \((18.98, 21.02)\). This confidence interval is based on a simple random sample of 36 observations. Calculate the sample mean and standard deviation. Assume that all conditions necessary for inference are satisfied. Use the \(t\)-distribution in any calculations.

5. A market researcher wants to evaluate car insurance savings at a competing company. Based on past studies he is assuming that the standard deviation \(\sigma\) of savings is $100. He wants to collect data such that he can get a margin of error of no more than $10 at a 95% confidence level. How large of a sample should he collect?

6. (MSSC) Nonparametric methods for estimating a population mean do not assume any specific distributional form (like normality) for the population. Instead, they rely on the data’s rank or resampling properties. Describe the idea of boostrapping, and how we can use boostrapping to construct a confidence interval for \(\mu\).

7. 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?