Statistics: Probability Distribution, Binomial, Random Samples, z-scores & Variation

1. Determine whether each of the distributions given below represents a probability distribution. Justify your answer.

(A) x 1 2 3 4
P(x) 1/12 5/12 1/3 1/12

(B)x 3 6 8
P(x) 2/10 .5 1/5

(C)x 20 35 40 50
P(x) 0.4 -0.2 0.5 0.3

2. Consider a binomial distribution with 14 identical trials and a probability of success of 0.4.
i. Find the probability that x = 3 using the binomial tables.
ii. Use the normal approximation to find the probability that x = 3. Show all work.

3. The diameters of oranges in a certain orchard are normally distributed with a mean of 4.85 inches and a standard deviation of 0.40 inches. Show all work.

(A) What percentage of the oranges in this orchard have diameters less than 6.3 inches?
(B) What percentage of the oranges in this orchard are larger than 4.75 inches?
(C) A random sample of 100 oranges is gathered and the mean diameter obtained was 4.75. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 4.75 inches?
(D) Why is the z-score used in answering (A), (B), and (C)?
(E) Why is the formula for z used in (C) different from that used in (A) and (B)?

4. Assume that the population of heights of male college students is approximately normally distributed with mean u of 68 inches and standard deviation sigma of 3.75 inches. A random sample of 16 heights is obtained. Show all work.

(A) Find the proportion of male college students whose height is greater than 70 inches.
(B) Find the mean and standard error of the distribution.
(C) Find P( > 70)

5. Answer the following questions regarding the normal, standard normal and binomial distributions.

(A) What conditions must be met in order to use the normal distribution to approximate the binomial distribution?
(B) How does the standard normal distribution differ from the normal distribution?
(C) Why is the correction for continuity necessary when the normal distribution is used to approximate a binomial distribution?

6. Four green marbles are selected, one at a time from a bin of marbles containing 6 black, 6 red and 6 green marbles. Let x represent the number of green marbles drawn in 4 draws.

(A) If this experiment is completed without replacing the marbles, explain why x is not a binomial random variable.
(B) If this experiment is completed with replacement of the marbles, explain why x is a binomial random variable.