Week 10 Quiz

A fair die has faces 1, 2, 3, 4, 5, 6, each with chance 1 ⁄ 6, so its population mean is μ = 3.5 points and its population standard deviation is σ = 1.7078 points. You roll it 30 times and compute the average of the faces, x̄. Which statement about μ and x̄ is correct?

Select one.

• μ = 3.5 is a parameter that stays fixed, while x̄ is a statistic that changes from one set of 30 rolls to the next
• x̄ is a parameter, because it describes the 30 rolls that were actually made
• μ and x̄ are both statistics, because both are averages of die faces
• μ changes every time the die is rolled again, while x̄ stays fixed once it is computed

Two classmates each roll the same fair die 30 times, following the same steps, and get different averages. This shows that at least one of them must have made a mistake.

True or false.

• True
• False

You average two rolls of a fair die. The average of two rolls is the sum of the two faces divided by 2, so the possible averages are 1, 1.5, 2, and so on up to 6, eleven values in all. What is the chance that the average is exactly 3.5?

Select one.

• 1 ⁄ 36
• 1 ⁄ 11
• 6 ⁄ 36
• 3 ⁄ 36

A fair die has faces 1, 2, 3, 4, 5, 6, each with chance 1 ⁄ 6, so its population mean is μ = 3.5 points and its population standard deviation is σ = 1.7078 points. You roll the die 30 times and average the faces. What is the standard error of that average, σ ⁄ √n? Enter the decimal to four places.

Type a short answer.

Suppose single commute times follow a normal model with μ = 32 minutes and σ = 6 minutes. What is the standard error of the average of n = 4 randomly chosen trips?

Select one.

• 6 minutes
• 3 minutes
• 1.5 minutes
• 24 minutes

A researcher draws many random samples of 100 purchases from a store where most purchases are small and a few are very large, so the population is strongly right-skewed. What does the Central Limit Theorem say about this situation?

Select one.

• The 100 purchases in each sample will form a bell shape, because 100 is more than 30
• Nothing useful, because the Central Limit Theorem applies only when the population follows a normal model
• The standard deviation of single purchases shrinks as each sample gets larger
• The sample means of 100 purchases follow an approximately normal model, even though each sample's purchases stay right-skewed

A fair die has faces 1, 2, 3, 4, 5, 6, each with chance 1 ⁄ 6, so its population mean is μ = 3.5 points and its population standard deviation is σ = 1.7078 points. For averages of 30 rolls, the standard error is 0.3118 points. Using two standard errors and the empirical rule, between which values do about 95% of 30-roll averages land?

Select one.

• Between 3.19 and 3.81
• Between 0.08 and 6.92
• Between 2.88 and 4.12
• Between 3.39 and 3.61

Suppose single commute times follow a normal model with μ = 32 minutes and σ = 6 minutes. The standard error of the average of 4 trips is 3 minutes. About what share of four-trip averages run longer than 38 minutes?

Select one.

• About 15.9%
• About 2.5%
• About 5%
• About 0.15%

Suppose 50% of a large population favors a proposal, so p = 0.5, and a pollster surveys n = 1,000 people at random. What is the standard deviation of the sample proportion p̂, √(p(1 − p) ⁄ n), rounded to four decimals?

Select one.

• 0.00025
• 0.5
• 0.0316
• 0.0158

For which of these statistics is a normal model for its sampling distribution reasonable under this course's rules? Select all that apply.

Select every option that applies.

• The sample proportion p̂ in a random poll of 1,000 people when p = 0.5, so np = 500 and n(1 − p) = 500
• The proportion of sixes in 12 rolls of a fair die, where np = 12 × 1 ⁄ 6 = 2
• The average of 4 trips when single commute times are assumed to follow a normal model with μ = 32 minutes and σ = 6 minutes
• The 30 individual faces in one sample of 30 rolls of a fair die
• The average of 30 rolls of a fair die, whose population of faces is flat and symmetric