# Explain the difference between a discrete and a continuous random variable.

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) 1/8 1/8 3/8 1/8

b) x 3 6 8

P (x) 0.2 0 1

c) x 20 30 40 50

P (x) 0.3 0.2 0.1 0.4

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 = 100 compare to the bell-shaped curve for the sampling distribution of sample means for samples of size n = 60?

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 15 identical trials, and a probability of success of 0.5

a. Find the probability that x = 2 using the binomial tables

b. Use the normal approximation to find the probability that x = 2

9) The diameters of oranges in a certain orchard are normally distributed with a mean of 5.26 inches and a standard deviation of 0.50 inches.

a. What percentage of the oranges in this orchard have diameters less than 4.5 inches?

b. What percentage of the oranges in this orchard is larger than 5.12 inches?

c. A random sample of 100 oranges is gathered and the mean diameter obtained was 5.12. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 5.12 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 male college students is approximately normally distributed with mean m of 68 inches and standard deviation s of 3.75 inches. A random sample of 16 heights is obtained.

a. Describe the distribution of x, height of the college student.

b. Find the proportion of male college students whose height is greater than 70 inches.

c. Describe the distribution of x, the mean of samples of size 16.

d. Find the mean and standard error of the distribution.

e. Find P ( > 70)

f. Find P ( < 67)

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#### Solution Preview

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

A discrete random variable takes in numerical (or integral) values, e.g. Binomial distribution, Poisson's distribution.

A continuous random variable can take in all real values, e.g., normal distribution, uniform distribution.

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

a) x 1 2 3 4

P(x) 1/8 1/8 3/8 1/8

∑ P(x) = 1/8 + 1/8 + 3/8 + 1/8 + 6/8 ≠ 1

So this is not a probability distribution

b) x 3 6 8

P(x) 0.2 0 1

∑ P(x) = 0.2 + 0 + 1 = 1.2 ≠ 1

So this is not a probability distribution

c) x 20 30 40 50

P(x) 0.3 0.2 0.1 0.4

∑ P(x) = 0.3 + 0.2 + 0.1 + 0.4 = 1

So this is a probability distribution

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.

This is because the probability of getting an ace will change from trial to trial, as the total number of cards in the deck changes.

b. If this experiment is completed with replacement, explain why x is a binomial random ...

#### Solution Summary

This solution provides answers to various questions involving probability including continuous random variables, probability distribution, binomial random variables, binomial distributions, and normal distributions.

A Discrete and A Continuous Random Variable

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

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

a)

x

1

2

3

4

P (x)

1/8

1/8

3/8

1/8

b)

x

3

6

8

P (x)

0.2

0

1

c)

x

20

30

40

50

P (x)

0.3

0.2

0.1

0.4

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.

If this experiment is completed without replacement, explain why x is not a binomial random variable.

If this experiment is completed with replacement, explain why x is a binomial random variable.

How does the bell-shaped curve for the sampling distribution of sample means for samples of size n = 100 compare to the bell-shaped curve for the sampling distribution of sample means for samples of size n = 60?

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.

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

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?

Consider a binomial distribution with 15 identical trials, and a probability of success of 0.5

Find the probability that x = 2 using the binomial tables

Use the normal approximation to find the probability that x = 2

The diameters of oranges in a certain orchard are normally distributed with a mean of 5.26 inches and a standard deviation of 0.50 inches.

What percentage of the oranges in this orchard have diameters less than 4.5 inches?

What percentage of the oranges in this orchard is larger than 5.12 inches?

A random sample of 100 oranges is gathered and the mean diameter obtained was 5.12. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 5.12 inches?

Why is the z-score used in answering (a), (b), and (c)?

Why is the formula for z used in (c) different from that used in (a) and (b)?

Assume that the population of heights of male college students is approximately normally distributed with mean m of 68 inches and standard deviation s of 3.75 inches. A random sample of 16 heights is obtained.

Describe the distribution of x, the height of the college student.

Find the proportion of male college students whose height is greater than 70 inches.

Describe the distribution of , the mean of samples of size 16.

Find the mean and standard error of the distribution.

Find P ( > 70)

Find P ( < 67)

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