Week 6 Quiz

This week uses four random variables: X, the face of one fair die; S, the sum of two fair dice; D, the number of days in a month picked at random from a common year; and B, the body mass of a penguin picked at random from the 342 measured birds. Which one is a continuous random variable?

Select one.

• X, the face of one fair die
• S, the sum of two fair dice
• B, the body mass of a penguin picked at random
• D, the number of days in a month picked at random

Someone miscounts the 31-day months and writes this table for D, the number of days in a month picked at random: P(D = 28) = 1 ⁄ 12, P(D = 30) = 4 ⁄ 12, and P(D = 31) = 8 ⁄ 12. Which statement is correct?

Select one.

• It is not legitimate: every entry is between 0 and 1, but the entries add to 13 ⁄ 12
• It is legitimate: every entry is between 0 and 1
• It is legitimate: the values 28, 30, and 31 are whole numbers
• It is not legitimate: an entry of 8 ⁄ 12 is too large to be a probability

X is the face of one fair die, with values 1, 2, 3, 4, 5, 6, each with probability 1 ⁄ 6. What is the expected value E(X)?

Select one.

• 21
• 6
• 3
• 3.5

For one fair die, E(X) = 3.5. True or false: this means a player should expect the next single roll to show about 3.5.

True or false.

• True
• False

S is the sum of two fair dice. Its values 2 through 12 happen in 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 of the 36 equally likely outcomes, and the products of each sum and its number of ways add to 252. What is the expected value E(S)? Enter a number.

Type a short answer.

For one fair die, the six squared deviations from the mean 3.5 are 6.25, 2.25, 0.25, 0.25, 2.25, and 6.25, which add to 17.5. Each face has probability 1 ⁄ 6. What is the variance Var(X)?

Select one.

• 17.5 ⁄ 5 = 3.5 square units
• 17.5 square units
• 35 ⁄ 12 ≈ 2.9167 square units
• √2.9167 ≈ 1.71 square units

S is the sum of two fair dice, with E(S) = 7 and SD(S) ≈ 2.4152. A player tracks S − 7, how far each total lands from 7. What are the mean and the standard deviation of S − 7?

Select one.

• Mean 7 and SD ≈ 2.4152
• Mean 0 and SD ≈ 2.4152
• Mean 0 and SD 0
• Mean 0 and SD ≈ 5.83

D is the number of days in a month picked at random from a common year, with E(D) = 365 ⁄ 12 days and SD(D) ≈ 0.86 days. The month's length in hours is H = 24 × D. Which pair gives E(H) and SD(H)?

Select one.

• E(H) = 730 hours and SD(H) ≈ 0.86 hours
• E(H) = 754 hours and SD(H) ≈ 24.86 hours
• E(H) ≈ 30.42 hours and SD(H) ≈ 20.69 hours
• E(H) = 730 hours and SD(H) ≈ 20.69 hours

A spinner's pointer is equally likely to stop anywhere on a dial marked from 0 to 6, so the density curve for its stopping point Y is flat at height 1 ⁄ 6. What is P(2 ≤ Y ≤ 3.5)?

Select one.

• 1 ⁄ 6
• 0.25
• 1.5
• 0

Smooth the histogram of the 342 penguin body masses into a density curve. For a penguin picked at random, the curve gives P(B = 4,000 g exactly) = 0. What does that zero mean?

Select one.

• A single point has no width, so it has no area, while a range near 4,000 g has real probability
• No penguin in the field log weighs anywhere near 4,000 g, because a probability of 0 means it can never happen
• The density curve has height 0 at 4,000 g
• A mass of 4,000 g lies outside the range of the 342 masses