Q: For a binomial random variable with $$n = 50$$ and $$p = 0.8$$, what is the standard deviation $$\sigma_X$$?

$$\sigma_X = \sqrt{50 \times 0.8 \times 0.2}$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that can only take on a finite or countable number of values.

Question 2

Suppose X is the number of cars passing through a toll booth in one hour. Which type of random variable is X?

Accepted Answer

Discrete, because you can count the number of cars.

Question 3

Which of the following is NOT a requirement for a discrete probability distribution?

Accepted Answer

The mean must be greater than the standard deviation.

Question 4

Given the probability distribution: $$x = 0, 1, 2, 3$$ with $$P(x) = 0.2, 0.5, 0.2, 0.1$$ respectively, what is the mean $$\mu_X of X$$?

Accepted Answer

$$\mu_X = 1.3$$

Question 5

The Law of Large Numbers states that as the number of trials increases, the sample mean:

Accepted Answer

Will approach the mean of the random variable.

Question 6

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of defective items in a batch of 10, where each item is either defective or not.

Question 7

If a binomial experiment has $$n = 20$$ and $$p = 0.3$$, what is the mean $$\mu_X$$?

Accepted Answer

$$\mu_X = 6$$

Question 8

For a binomial random variable with $$n = 50$$ and $$p = 0.8$$, what is the standard deviation $$\sigma_X$$?

Accepted Answer

$$\sigma_X = \sqrt{50 	imes 0.8 	imes 0.2}$$

Question 9

A Poisson process is appropriate when:

Accepted Answer

The number of successes in non-overlapping intervals are independent.

Question 10

If the average number of emails received per hour is 5, what is the probability of receiving exactly 3 emails in one hour according to the Poisson distribution?

Accepted Answer

$$P(3) = \frac{5^3$}{3!} e$$^{-5}$$

Question 11

Which of the following statements about the shape of a binomial distribution is TRUE as $$n$$ increases and $$p is $$not too close to 0 or 1?

Accepted Answer

The distribution becomes approximately bell-shaped.

Question 12

A probability histogram for a binomial distribution with $$n = 8$$ and $$p = 0.85$$ will most likely be:

Accepted Answer

Skewed to the left.

Question 13

If $$\mu_X = 10$$ and $$\sigma_X = 2$$, according to the Empirical Rule, which interval contains about 95% of the observations?

Accepted Answer

$$[6, 14]$$

Question 14

A researcher finds 162 successes in a binomial experiment where $$n = 400$$, $$p = 0.35$$, $$\mu_X = 140$$, and $$\sigma_X = 9.54. Is $$this result unusual?

Accepted Answer

Yes, because 162 is greater than $$\mu_X + 2\sigma_X.$$

Question 15

For a Poisson random variable with $$\lambda = 0.6$$ per gram, what is the mean number of occurrences in a 5-gram sample?

Accepted Answer

$$\mu_X = 3$$

Question 16

Which formula gives the standard deviation of a Poisson random variable with mean $$\mu_X$$?

Accepted Answer

$$\sigma_X = \sqrt{\mu_X}$$

Question 17

A probability distribution is given by $$P(x) = 0.2$$ for $$x=0$$, $$P(x) = 0.5$$ for $$x=1$$, $$P(x) = 0.3$$ for $$x=2. Is $$this a valid probability distribution?

Accepted Answer

Yes, because all probabilities are between 0 and 1 and sum to 1.

Question 18

Which of the following is a real-world example of a Poisson process?

Accepted Answer

Counting the number of emails received per hour.

Question 19

If $$P(x) = \frac{\mu^x}{x!} e$$^{-\mu}$$, what does $$\mu$$ represent in the Poisson distribution?

Accepted Answer

The average number of occurrences in the interval.

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