Random Variables, Probability Distributions

Please see the attached file for the fully formatted problems.

1. Explain the difference between a discrete and a continuous random variable. Give two examples of each type of random variable.

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

a)
x 1 2 3 4
P(x) 1/4 5/12 1/3 1/6

b)
x 3 6 8
P(x) 0.1 3/5 0.3

c)
x 20 30 40 50
P(x) 0.2 -0.2 0.7 0.3

3. Four cards are selected, one at a time from a standard deck of 52 cards. Let x represent the number of aces drawn in a set of 4 cards.
a. If this experiment is completed without replacement, explain why x is not a binomial random variable.
b. If this experiment is completed with replacement, explain why x is a binomial random variable.

4. How does the bell-shaped curve for the sampling distribution of sample means for samples of size n = 120 compare to the bell-shaped curve for the sampling distribution of sample means for samples of size n = 95?

5. What are the characteristics of the normal distribution? Why is the normal distribution important in statistical analysis? Provide an example of an application of the normal distribution.

6. In your own words describe the standard normal distribution. Explain why it can be used to find probabilities for all normal distributions.

7. Explain why the normal distribution can be used as an approximation to the binomial distribution. What conditions must be met to use the normal distribution to approximate the binomial distribution? Why is a correction for continuity necessary?

8. 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

9. 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.

a) What percentage of the oranges in this orchard have diameters less than 4.3 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)?

10. Assume that the population of heights of female college students is approximately normally distributed with mean  of 65 inches and standard deviation  of 2.75 inches. A random sample of 15 heights is obtained.

a) Describe the distribution of x, height of the college student
b) Find the proportion of female college students whose height is greater than 67 inches.
c) Describe the distribution of , the mean of samples of size 15.
d) Find the mean and standard error of the distribution
e) Find P ( > 67)
f) Find P ( < 64)