Question 1

Which of the following best describes a random variable that can take all natural numbers (1, 2, 3, 4, ...)?

Accepted Answer

It is a discrete random variable.

Question 2

If a random variable is approximately normal, which of the following is true?

Accepted Answer

The ratio of interquartile range and standard deviation is approximately 1.35.

Question 3

Which two statements correctly describe sample distribution and sampling distribution?

Accepted Answer

The sampling distribution shows the distribution of a statistic (like the mean) across many samples from a population, and the sample distribution shows the distribution of values within a single sample from a population.

Question 4

Given the three histograms (left: red, middle: green, right: blue), which one represents the distribution of the population, the sample mean when $$n=2$$, and the sample mean when $$n=30$$? What statistical principle explains the difference in their shapes?

Accepted Answer

Left: population, Middle: sample mean $$n=2$$, Right: sample mean $$n=30$$; Central Limit Theorem.

Question 5

If $$P(A) = 0.4$$ and $$P(B) = 0.3$$, can events A and B be mutually exclusive? Why or why not?

Accepted Answer

No, because $$P(A \cap B)$$ would have to be zero, but their probabilities do not sum to more than 1.

Question 6

If a random variable $$X is $$uniformly distributed from 4 to 10, what are its mean and standard deviation?

Accepted Answer

Mean: $$\frac{4+10}{2} = 7$$, Standard deviation: $$\sqrt{\frac{(10-4)^2$}{12}} = 1.732$$

Question 7

At a city intersection, the number of stop sign tickets in a day follows a Poisson distribution with $$\lambda=4. $$What is the probability that there are at least 6 tickets in a day?

Accepted Answer

$$1 - \sum_{k=0}^{5} \frac{e$$^{-4}$$ 4^k}{k!}$$

Question 8

If the time between stop sign tickets follows an exponential distribution with $$	heta=6$$ hours, what is the probability that another ticket will be issued within 5 hours?

Accepted Answer

$$1 - e$$^{-5/6}$$

Question 9

At a coffee house, 80% of buyers choose regular coffee. Out of the next 5 customers, what is the probability that exactly 3 will choose regular coffee?

Accepted Answer

$$\binom{5}{3} (0.8)^3$ (0.2)^2$

Question 10

Out of the next 20 coffee customers, what is the probability that at least 13 will choose regular coffee?

Accepted Answer

$$\sum_{k=13}^{20} \binom{20}{k} (0.8)^k (0.2$$)^{20-k}$$

Question 11

A box contains 4 dimes (10 cents) and 2 nickels (5 cents). If 3 coins are randomly selected without replacement, what is the probability that the selected coins total 20 cents?

Accepted Answer

$$\frac{\binom{4}{2} \binom{2}{1}}{\binom{6}{3}}$$

Question 12

In a Venn diagram showing 9th grade students in clubs B and C, if 17 students are in club B but not C, and one student is randomly selected, what is the probability that the student is in both clubs?

Accepted Answer

$$\frac{n(B \cap C)}{n(	ext{Total})}$$

Question 13

Suppose men’s neck sizes are normally distributed with mean 15.7 inches and standard deviation 0.7 inch. What is the proportion of men whose shirt size is M (15 ≤ Size \u003c 16)?

Accepted Answer

$$P(15 \leq X \u003c 16) = P\left(\frac{15-15.7}{0.7} \leq Z \u003c \frac{16-15.7}{0.7}\right)$$

Question 14

If you play American roulette (18 red, 18 black, 2 green) and bet $$1 on red, what is the expected value of your net gain X$ per play?

Accepted Answer

$$E[X] = 1 	imes \frac{18}{38} + (-1) 	imes \frac{20}{38}$$

Question 15

If you play American roulette 80 times, how much money would you expect to win or lose?

Accepted Answer

$$80 	imes E[X] = 80 	imes \left(\frac{18}{38} - \frac{20}{38}ight)$$

Question 16

If you play American roulette 80 times, what is the probability that you have a positive gain (average net gain $$\u003e0$$)?

Accepted Answer

$$P\left(\bar{X} \u003e 0ight) = P\left(Z \u003e \frac{0 - E[X]}{\sqrt{\frac{	ext{Var}(X)}{80}}}ight)$$

Midterm Exam 2 Study Guide – STAT211 Statistics I

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