ACT Math: Statistics and Probability (Drill 3)

Question 1. The frequency table below shows the number of hours per week that 30 students spend studying.
Hours per week    Students
0, 2                   4
3, 5                   9
6, 8                  11
9, 11                  6
What fraction of students study 6 or more hours per week?

• A) 11/30
• B) 17/30 ✓
• C) 3/5
• D) 1/2

Explanation: Students studying 6 or more hours per week fall in the 6–8 and 9–11 categories: 11 + 6 = 17. Total students = 30. Fraction = 17/30. Choice A counts only the 6–8 group (11) and ignores the 9–11 group. Choice C gives 18/30 = 3/5, adding 11 + 7 instead of 11 + 6, a reading error in the 9–11 row. Choice D gives 15/30 = 1/2, perhaps from computing 9 + 6 = 15 and using the 3–5 group count instead of the 6–8 group count.

Question 2. In a class, the final grade is computed as follows: homework counts 20%, quizzes count 30%, and the final exam counts 50%. A student earns an 80 on homework, a 70 on quizzes, and a 90 on the final exam. What is the student's final grade?

• A) 80
• B) 82 ✓
• C) 83
• D) 85

Explanation: Weighted average = (0.20)(80) + (0.30)(70) + (0.50)(90) = 16 + 21 + 45 = 82. Choice A results from computing the simple (unweighted) average: (80 + 70 + 90) ÷ 3 = 80. Choice C results from a small arithmetic error in one of the components, such as computing (0.30)(70) = 22 instead of 21: 16 + 22 + 45 = 83. Choice D results from computing the weighted average with equal weights of 1/3 each but applying the 50% weight to the highest score twice: (80 + 70 + 90 + 90) ÷ 4 = 330 ÷ 4 = 82.5 ≈ 85 via rounding error.

Question 3. A square dartboard has a side length of 20 inches. A circular target is painted in the center of the board with a radius of 5 inches. Assuming a dart lands at a uniformly random point on the board, what is the probability that it lands inside the circular target? (Use π ≈ 3.14)

• A) approximately 0.25
• B) approximately 0.196 ✓
• C) approximately 0.785
• D) approximately 0.0491

Explanation: Geometric probability = favorable area ÷ total area. Area of circle = π(5)² = 25π ≈ 78.5 in². Area of square = 20² = 400 in². Probability = 78.5 / 400 ≈ 0.196. Choice A (0.25) results from computing the ratio of the radius to the side length: 5/20 = 0.25, treating this as a linear ratio rather than an area ratio. Choice C (0.785) results from computing π(5)² / π(10)² = 25/100 = 0.25... no, 0.785 ≈ π/4, which is the probability if the circle's diameter equaled the board's side (radius = 10): π(10)² / 400 = 314/400 = 0.785. Choice D results from computing the circle area using the diameter (10) instead of the radius (5), then dividing by the wrong total: π(5)² / 400² ≈ 0.049.

Question 4. A drawer contains 6 black socks and 4 white socks. Two socks are drawn one at a time without replacement. What is the probability that both socks are black?

• A) 9/25
• B) 1/3 ✓
• C) 36/100
• D) 8/15

Explanation: P(first sock is black) = 6/10. After drawing one black sock, 5 black socks and 4 white socks remain (9 total). P(second sock is black | first was black) = 5/9. P(both black) = (6/10)(5/9) = 30/90 = 1/3. Choice A results from treating the draws as independent (with replacement): (6/10)² = 36/100 = 9/25. Choice C is the same error as A expressed as 36/100 before simplifying. Choice D results from computing (6/10)(5/9) but simplifying to 30/90 as 8/15 via an arithmetic error, or from computing (6 + 5) / (10 + 9) = 11/19 and misreading as 8/15.

Question 5. A teacher records scores on a 50-point quiz for 20 students. The scores are summarized below.
Score Range    Students
10, 19                1
20, 29                2
30, 39                5
40, 49                9
50                      3
Which of the following best describes this distribution?

• A) Symmetric, with most scores near the center
• B) Skewed left, with most scores concentrated at the low end based on the given information
• C) Skewed right, with a long tail toward higher scores
• D) Skewed left, with most scores concentrated at the high end ✓

Explanation: The frequencies increase as scores increase: 1, 2, 5, 9, 3. The bulk of the data (12 of 20 students) is in the 40–50 range, with a long tail stretching toward the lower scores. This is a left-skewed (negatively skewed) distribution, the tail points to the left (toward lower values) and most scores are concentrated at the high end. Choice A is incorrect: a symmetric distribution would have roughly equal frequencies above and below the center. Choice B confuses the direction: "skewed left" is correct but "concentrated at the low end" is backwards, in a left-skewed distribution, the bulk of the data is at the HIGH end, with the tail pointing left. Choice C describes a right-skewed distribution, in which the tail points right and most scores cluster at the low end, the opposite of what we see here.