Homework 7

Inference about Variances

Important

Due Friday, Nov 7, 11:59 PM

• 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.

• In your homework, please number questions in order.

• 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.

Homework Questions

1. The data about male and female pulse rates are summarized below.
   1. Construct a 95% CI for \(\sigma_{male}\) of pulse rates for males.
   2. Construct a 95% CI for \(\sigma_{male}/\sigma_{female}\).
   3. Does it appear that the population standard deviations for males and females are different? Why or why not?

Pulse Rate by Gender
	Male	Female
Mean (x̄)	70	76
Standard deviation (s)	10	13
Sample size (n)	16	11

2. A production process for filling orange juice containers labeled as 64 ounces is monitored for the actual amount of juice in the container. The process is designed such that the amount of juice in the containers has a normal distribution with a mean of 64.3 ounces and a standard deviation of .15 ounces. The process is monitored by randomly selecting 24 containers every hour and measuring the actual amount of juice in the containers. An increase in the standard deviation beyond .15 ounces with the mean remaining at 64.3 ounces will result in a production run with too many underfilled and overfilled containers. The following data are the actual amounts of juice in a random sample of 24 containers.
   1. If the amount of juice in the containers has a normal distribution with a mean of 64.3 ounces and a standard deviation of .15 ounces, what proportion of containers filled on the production line will be underfilled (contain less than 64 ounces)?
   2. Construct a 95% confidence interval on the process standard deviation.
   3. Do the test on the claim that the process standard deviation is greater than .15 ounces. Use \(\alpha = 0.05\).
   4. What is the p-value of your test in (c)? (We can use pchisq())
   5. If you do (b) and (c) correctly, you should find that (b) and (c) lead to a different conclusion about the claim that \(\sigma > 0.15\). Can you provide the reasoning?

64.37	64.26	64.22	64.42	64.13	64.44	64.64	64.19	63.85	64.17	64.21	64.23
64.64	64.12	63.98	64.34	64.20	64.31	64.15	64.09	64.33	64.19	64.57	64.19

3. 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?