Confidence Intervals and Probability Distributions

Hey you guys we were learning about probability distributions and confidence intervals and I was wondering if you guys could help me answer these questions as well as try and explain how you've come to the answer so I can get a better understanding,

1. I want to know how my grade on the midterm compares to the rest of my classmates. My
professor tells me that the grades are normally distributed, there was an average score of 83
and a standard deviation of 8. What percentage of students received a higher score on the
midterm than me if I scored a 92? What did the top 15% of students earn? What did the bottom
15% of students earn?

2. A local hospital is examining the length of hospital stays for its patients. They find that the
length of stays for their patients is normally distributed. The mean hospital stay is 200 hours and
the standard deviation is 75 hours. What percentage of patients had a hospital stay longer than
300 hours? What percentage of patients had a stay less than 75 hours? What length of stay (in
hours) would a patient need to be considered in the top 25%? What would the length of the
stay need to be (in hours) to be in the bottom 10%?

3. In our random sample of 50 Iowa residents, we find an average of 10 connected devices in
each home (TV’s, computers, cell phones, lights, etc.). The standard deviation of all Iowa
residents is 2. Estimate the population mean using a 99% confidence interval.

4. A researcher interested in the age people begin smoking knows that the standard deviation for
United States smokers is 1.75 years, but he does not know the average age of onset. In order to
estimate the average age of onset, he randomly selects 35 smokers who have an average onset
age of 15. Using a 95% confidence interval, estimate the national average age of onset for