Week 7 Quiz

Which of these counts fits the binomial setting: a fixed number of trials, independent trials, two outcomes per trial, and the same chance of success on every trial?

Select one.

• Drawing 3 tickets one at a time from a jar of 10, keeping each ticket, and counting the winners
• Phoning customers until the first one answers, and counting the calls made
• Guessing on 10 four-option questions and counting the correct answers
• Recording each survey answer as agree, neutral, or disagree, and counting how many people gave each of the three answers

A student guesses at random on a 10-question multiple-choice quiz with four options per question, so each guess is right with chance p = 0.25 and X, the number of correct answers, is binomial with n = 10. Using the binomial probability formula, what is P(X = 1), rounded to four decimals?

Select one.

• 0.1877
• 0.0188
• 0.2500
• 0.0563

A student guesses at random on a 10-question multiple-choice quiz with four options per question, so each guess is right with chance p = 0.25 and X, the number of correct answers, is binomial with n = 10. The binomial table gives P(X = 0) = 0.0563, P(X = 1) = 0.1877, and P(X = 2) = 0.2816. What is P(X ≤ 2)? Enter the decimal to four places.

Type a short answer.

A student guesses at random on a 10-question multiple-choice quiz with four options per question, so each guess is right with chance p = 0.25 and X, the number of correct answers, is binomial with n = 10. A spreadsheet's =BINOM.DIST(4, 10, 0.25, TRUE) returns P(X ≤ 4) = 0.9219, and =BINOM.DIST(5, 10, 0.25, TRUE) returns P(X ≤ 5) = 0.9803. What is P(X ≥ 5), the chance of scoring 5 or more?

Select one.

• 0.0197
• 0.9219
• 0.0584
• 0.0781

A student guesses at random on a 10-question multiple-choice quiz with four options per question, so each guess is right with chance p = 0.25 and X, the number of correct answers, is binomial with n = 10. What are the binomial mean μ and standard deviation σ of X?

Select one.

• μ = 2.5 and σ ≈ 1.875
• μ = 2.5 and σ ≈ 1.3693
• μ = 7.5 and σ ≈ 1.3693
• μ = 2 and σ ≈ 1.3693

A student guesses at random on a 10-question multiple-choice quiz with four options per question, so each guess is right with chance p = 0.25 and X, the number of correct answers, is binomial with n = 10. The binomial mean is μ = np = 2.5. True or false: this means one guesser should expect to score exactly 2.5 on the quiz.

True or false.

• True
• False

A student guesses at random on a 10-question multiple-choice quiz with four options per question, so each guess is right with chance p = 0.25 and X, the number of correct answers, is binomial with n = 10. Using this course's large-counts rule of thumb, may a normal curve stand in for the distribution of X?

Select one.

• Yes, because n = 10 questions is at least 10
• Yes, because the mean 2.5 is smaller than n
• No, because np = 2.5 is below 10
• No, because p = 0.25 is below 0.5

Now the same guessing happens on a 100-question quiz with four options per question, so n = 100 and p = 0.25. Which statement is correct?

Select one.

• np = 25 and n(1 − p) = 75 are both at least 10, so a normal curve with mean 25 and standard deviation about 4.3301 is a reasonable stand-in
• np = 25 and n(1 − p) = 75 are both at least 10, so a normal curve with mean 25 and standard deviation about 18.75 is a reasonable stand-in
• np = 2.5 is below 10, so a normal curve is still not a reasonable stand-in for the count of correct answers
• n(1 − p) = 75 is at least 10, but np = 25 is not large enough, so a normal curve is not a reasonable stand-in for the count of correct answers here

Select every statement that is true of a normal curve as this course uses it.

Select every option that applies.

• It is set by two numbers: its mean μ, which places the center, and its standard deviation σ, which sets the spread
• The area under the curve over a range of values is the probability of landing in that range
• A set of real measurements may be called normal as a fact, without stating it as a model or checking it
• Its long, thin tail always runs to the right, like the 10-question guessing distribution
• The total area under the whole curve is 1

For the 10-question guessing quiz, the binomial table rises from k = 0 to its peak at k = 2 and then trails off toward the high counts, with k = 9 and k = 10 rounding to 0.0000. For the 100-question version, the bars center on 25. Which description of the two shapes is correct?

Select one.

• Left-skewed at 10 questions, because the pile is on the left; a nearly symmetric bell at 100 questions
• Right-skewed at both lengths, because a pure guesser always scores low no matter how many questions the quiz has
• Symmetric at both lengths, because p stays 0.25 on every question
• Right-skewed at 10 questions, because the long tail runs to the right; a nearly symmetric bell at 100 questions