Another sampling distributions question?! 10 points promise!?

Question

67% of vehicles on the roads during rush hour are occupied by just the driver.

a) If you choose 380 vehicles at random, the mean number of vehicles that carry just the driver is ________ and the standard deviation is ________

b) If you choose 380 vehicles at random, the mean proportion of vehicles that carry just the driver is ________ and the standard deviation is ________


Please and thank you =)

Answer 1

Let X be the number of vehicles with just the driver. X has the binomial distribution with n = 380 trials and success probability p = 0.67.

In general, if X has the binomial distribution with n trials and a success probability of p then
P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)
for values of x = 0, 1, 2, ..., n
P[X = x] = 0 for any other value of x.

this is found by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.
Or, to be more accurate, the binomial is the sum of n independent and identically distributed Bernoulli trials.

the mean of the binomial distribution is n * p
the variance of the binomial distribution is n * p * (1 - p)


a)
mean is n * p = 380 * 0.67 = 254.6
variance is n * p * ( 1- p) = 380 * 0.67 * ( 1 - 0.67 ) = 84.018
the standard deviation is sqrt( variance) = sqrt(84.018) = 9.166133


b) the mean, or expectation of p, the proportion, is:
E(p) = E(X/380) = E(X) / 380 = 254.6 / 380 = 0.67

the variance of p is:
Var(p) = Var(X / 380) = (1/380)^2 * Var(X) = 84.018 / (380^2)
= 0.0005818421

the std dev of p is: sqrt(0.0005818421) = 0.02412140

Answer 2

a.) expected value of xbar= .67x380= 254.60
standard deviation of xbar: sigma xbar =sigma/ (square root of n) ??? i don't remember how to do this one. I think you would need the need the population standard deviation, sigma.

b) mean proportion: p=.67
standard deviation= square root of ( (.67(1-.67))/380)=.05422

Answer 3

n= 380 and p=.67 Let q = 1-p = .33

a) mean = np
std dev = sqrt(npq)

b) mean = p
std dev = sqrt(pq/n)