Question 1

What is the variance of the distribution of the average an draw of \(n\) observations from a population with mean \(\mu\) and variance \(\sigma^2\)

Answer 1

\[ \frac{\sigma^2}{n} \]

Question 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?

Answer 2

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

Question 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?

Answer 3

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

Question 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?

Answer 4

qnorm(0.95, 1100, 7.5)
## [1] 1112.336

Question 5

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

Answer 5

1 - pbinom(3,5,1/2)
## [1] 0.1875

Question 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?

Answer 6

pnorm(16, 15, 1) - pnorm(14, 15, 1)
## [1] 0.6826895

Question 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?

Answer 7

\[ \mathbb{E}\left[ \bar{X} \right] = \mu = 0.5 \]

mean(runif(1000))
## [1] 0.4940235

Question 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?

Answer 8

lambda = 5
ppois(10,3*lambda)
## [1] 0.1184644