Quiz 6

Solutions: 1. Identify the following densities of the normal distribution: a. Within one standard deviation of the mean. 68% (10 points) b. Within two standard deviations of the mean. 95% (10 points) c. Within three standard deviations of the mean. 99.7% (10 points) 2. Suppose the random variable x is best described by a normal distribution with μ = 25 and σ = 5. Find the z-score that corresponds to each of the following x values: a. x = 25 z = x − μ σ = 25 − 25 5 = 0 (10 points, partial credit of 5 points if they show they’re trying to find a z-score) b. x = 30 z = 30 − 25 5 = 1 (10 points, partial credit of 5 points if they show they’re trying to find a z-score) c. x = 10 z = 10 − 25 5 =− 3 (10 points, partial credit of 5 points if they show they’re trying to find a z-score) 3. Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 4. a. Find f(x). f ( x ) = 1 d − c = 1 4 − 2 = 1 2 (10 points, partial credit of 5 points if they show they’re trying to find a uniform density) b. Find the mean and standard deviation. μ = c + d 2 = 2 + 4 2 = 3 σ = d − c √ 12 = 4 − 2 √ 12 = .5774 (10 points, partial credit of 5 points if they show they’re trying to find a uniform density) c. Find P ( x > 2.78 ) . P ( x > 2.78 ) = P ( 2.78 < x < 4 ) = 4 − 2.78 4 − 2 = .61 (10 points, partial credit of 5 points if they show they’re trying to find a uniform density) 4. Suppose the random variable x has an exponential probability distribution with θ = 1. Find the mean and standard deviation of x . θ = μ = σ = 1 (10 points, partial credit of 5 points if they mess up and find λ=1/θ) (Note that this will appear as two questions online -each worth 5 points).