Please help these statistic problems!

1. Suppose the body mass index (BMI) for a large sample of patients has a mean of 23 and a standard deviation of 2. Assuming the distribution is normal (Gaussian), sketch what it might look like. (a) Lable your axes. (b) Indicate where the BMI equals its mean. (c) Indicate where the BMI is one standard deviation above and below the mean. (c) Now suppose you have a new sample, also normally distributed, with the same mean but a of only 1. Sketch that distribution in a way that makes clear the difference that a smaller makes.

2. In the problem above, using = 23 and = 2, what is the range that encompasses 95% of the observations and is symmetrical around the mean?

3. Suppose your organization conducts a weekly survey of 100 respondents, asking them about their sleep patterns the previous night. Formerly it was the case that each of them had a 5% chance of insomnia on a given night, but now it is 10%. What is the corresponding change in the standard deviation of reported insomnia totals you will experience, assuming the outcomes are indendent of one another? (That is, none of them live together, keeping each other awake....)

4. The amount of a virus in an individual's body is measured in virions and can be estimated by taking tissue samples. Suppose you have virion data for 50 unrelated individuals all infected with Covid; they don't know each other and don't physically interact at work or in the community. One of your variables is the number of virions they generate in a blood test. What type of distribution would you anticipate for the number of virions across this sample? Why?