1. What is the variance of the distribution of the average an IID draw of n observations from a population with mean mu and variance theta^2.

2. Suppose that diastolic blood pressures (DBPs) for men aged 35-44 are normally distributed with a mean of 80 (mm Hg) and a standard deviation of 10. About what is the probability that a random 35-44 year old has a DBP less than 70?

pnorm(70, mean=80, sd=10) 
## [1] 0.1586553

3. Brain volume for adult women is normally distributed with a mean of about 1,100 cc for women with a standard deviation of 75 cc. What brain volume represents the 95th percentile?

qnorm(0.95, mean=1100, sd=75)
## [1] 1223.364

4. Refer to the previous question. Brain volume for adult women is about 1,100 cc for women with a standard deviation of 75 cc. Consider the sample mean of 100 random adult women from this population. What is the 95th percentile of the distribution of that sample mean?

# When the distribution of the population is normal, then the distribution of the sample mean is also normal. For a normal population distribution with mean mu and standard deviation theta, the distribution of the sample mean is normal, with mean mu and standard deviation theta/sqrt(n)

qnorm(0.95, mean=1100, sd=75/10)
## [1] 1112.336

5. You flip a fair coin 5 times, about what’s the probability of getting 4 or 5 heads?

# all possiblities 
pall <- 2**5

# 4 heads with 5 flips
p1 <- (5*4*3*2)/(4*3*2*1)/pall

# 5 heads with 5 flips
p2 <- (5*4*3*2*1)/(5*4*3*2*1)/pall

p1+p2
## [1] 0.1875

6. The respiratory disturbance index (RDI), a measure of sleep disturbance, for a specific population has a mean of 15 (sleep events per hour) and a standard deviation of 10. They are not normally distributed. Give your best estimate of the probability that a sample mean RDI of 100 people is between 14 and 16 events per hour?

n<-100
mean <- 15
sd <- 10
SD <- sd/sqrt(n)
pnorm(16, mean = mean, sd = SD) - pnorm(14, mean = mean, sd = SD)
## [1] 0.6826895

7. Consider a standard uniform density. The mean for this density is .5 and the variance is 1 / 12. You sample 1,000 observations from this distribution and take the sample mean, what value would you expect it to be near?

quantile <- 0.5
mu <- 0.5
theta <- 1/12
n <- 1000 
SE <- theta/sqrt(n)
qnorm(quantile, mean = mu, sd = SE)
## [1] 0.5

8. The number of people showing up at a bus stop is assumed to be Poisson with a mean of 5 people per hour. You watch the bus stop for 3 hours. About what’s the probability of viewing 10 or fewer people?

ppois(10, lambda = 15)
## [1] 0.1184644