Normal Distribution

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Answer Key & Explanations

Normal Distribution · Grade 11 · Worksheet 1

1. Normal: μ=68, σ=4. What % between 60 and 76? Answer: 95 Solution: Calculate how many standard deviations 60 is from the mean: (60 - 68)/4 = -8/4 = -2 standard deviations Calculate how many standard deviations 76 is from the mean: (76 - 68)/4 = 8/4 = 2 standard deviations According to the 68-95-99.7 rule for normal distributions, approximately 95% of data falls…
   Full step-by-step solution

   Step 1: Calculate how many standard deviations 60 is from the mean: (60 - 68)/4 = -8/4 = -2 standard deviations Step 2: Calculate how many standard deviations 76 is from the mean: (76 - 68)/4 = 8/4 = 2 standard deviations Step 3: According to the 68-95-99.7 rule for normal distributions, approximately 95% of data falls within 2 standard deviations of the mean Step 4: Therefore, approximately 95% of values fall between 60 and 76 The answer is 95.

2. Charlotte is analyzing the lifespans of a particular model of LED light bulb produced by her company. The lifespans (in hours) are normally distributed with a mean of 12,000 hours and a standard deviation of 2,000 hours. The company's marketing department wants to advertise a 'long-life' warranty that covers the top 2.5% of bulbs. What is the minimum number of hours a bulb must last to qualify for this 'long-life' warranty? Answer: 15920 Solution: Identify the given information. Mean (μ) = 12,000 hours Standard deviation (σ) = 2,000 hours Top 2.5% means 97.5% of bulbs have a lifespan below this threshold. Use the 68-95-99.7 rule.
   Full step-by-step solution

   Step 1: Identify the given information. Mean (μ) = 12,000 hours Standard deviation (σ) = 2,000 hours Top 2.5% means 97.5% of bulbs have a lifespan below this threshold. Step 2: Use the 68-95-99.7 rule. The rule states that 95% of data lies within 2 standard deviations of the mean. This means 5% lies outside this range, with 2.5% in each tail. The top 2.5% tail starts at approximately 2 standard deviations above the mean. Step 3: Calculate the cutoff value. Value = μ + 2σ Value = 12,000 + 2(2,000) Value = 12,000 + 4,000 Value = 16,000 hours Step 4: For more precision, note that the 97.5th percentile corresponds exactly to a z-score of 1.96. Using this more precise value: Value = μ + 1.96σ Value = 12,000 + 1.96(2,000) Value = 12,000 + 3,920 Value = 15,920 hours The minimum lifespan for the 'long-life' warranty is 15,920 hours.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the area of this circumscribed circle? Answer: 25π Solution: Identify the triangle's vertices and sides. The vertices are A(0,0), B(6,0), and C(6,8). Side AB is from (0,0) to (6,0) → length = 6.
   Full step-by-step solution

   Step 1: Identify the triangle's vertices and sides. The vertices are A(0,0), B(6,0), and C(6,8). Side AB is from (0,0) to (6,0) → length = 6. Side BC is from (6,0) to (6,8) → length = 8. Side AC is from (0,0) to (6,8) → length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10. Step 2: Recognize the triangle is right-angled. Since AB is horizontal and BC is vertical, angle B is 90 degrees. Step 3: Find the circumradius of a right triangle. For a right triangle, the hypotenuse is the diameter of the circumscribed circle. Here, AC is the hypotenuse, length 10. So the diameter of the circle = 10. Step 4: Calculate the radius. Radius r = diameter / 2 = 10 / 2 = 5. Step 5: Find the area of the circle. Area = π * r^2 = π * 5^2 = π * 25 = 25π. Final answer: 25π

4. Hana is a botanist studying the heights of a particular species of tree in a forest reserve. The heights follow a normal distribution with a mean of 12.4 meters and a standard deviation of 2.1 meters. Hana selects a tree at random. What is the probability, expressed as a percentage, that the tree's height is between 10.3 meters and 14.5 meters? Use the 68-95-99.7 rule (empirical rule) to approximate your answer. Answer: 68% Solution: Calculate the z-score for 10.3 m. z = (10.3 - 12.4) / 2.1 = (-2.1) / 2.1 = -1. Calculate the z-score for 14.5 m.
   Full step-by-step solution

   Step 1: Calculate the z-score for 10.3 m. z = (10.3 - 12.4) / 2.1 = (-2.1) / 2.1 = -1. Step 2: Calculate the z-score for 14.5 m. z = (14.5 - 12.4) / 2.1 = 2.1 / 2.1 = 1. Step 3: Both boundaries are exactly 1 standard deviation from the mean (z = -1 and z = 1). Step 4: According to the 68-95-99.7 rule, approximately 68% of the data in a normal distribution lies within 1 standard deviation of the mean. Step 5: Therefore, the probability that a randomly selected tree has a height between 10.3 m and 14.5 m is approximately 68%. The answer is 68%.

5. Tane is analyzing the battery life of a new smartphone model. The battery life follows a normal distribution with a mean of 48 hours and a standard deviation of 6 hours. The manufacturer guarantees that the battery will last at least 36 hours. What percentage of smartphones are expected to have a battery life less than the guaranteed minimum of 36 hours? Answer: Approximately 2.5% Solution: Calculate how many standard deviations 36 hours is from the mean. z = (36 - 48) / 6 = -12 / 6 = -2 Use the 68-95-99.7 rule. A z-score of -2 means 36 hours is 2 standard deviations below the mean.
   Full step-by-step solution

   Step 1: Calculate how many standard deviations 36 hours is from the mean. z = (36 - 48) / 6 = -12 / 6 = -2 Step 2: Use the 68-95-99.7 rule. A z-score of -2 means 36 hours is 2 standard deviations below the mean. Step 3: According to the rule, 95% of data falls within 2 standard deviations of the mean (between 36 and 60 hours). This means 5% of data falls outside this range (below 36 hours and above 60 hours). Step 4: Since the normal distribution is symmetric, half of that 5% (2.5%) falls below 36 hours and half (2.5%) falls above 60 hours. Final Answer: Approximately 2.5% of smartphones are expected to have a battery life less than 36 hours.

6. Emma is analyzing the test scores from her Grade 11 mathematics class. The scores follow a normal distribution with a mean of 70 and a standard deviation of 10. Emma wants to know what percentage of students scored between 60 and 80. Use the 68-95-99.7 rule to determine this percentage. Answer: 68% Solution: Identify the mean and standard deviation. Mean (mu) = 70, Standard deviation (sigma) = 10. Determine how many standard deviations 60 and 80 are from the mean.
   Full step-by-step solution

   Step 1: Identify the mean and standard deviation. Mean (mu) = 70, Standard deviation (sigma) = 10. Step 2: Determine how many standard deviations 60 and 80 are from the mean. 60 is 10 below 70, so it is one standard deviation below the mean. 80 is 10 above 70, so it is one standard deviation above the mean. Step 3: The 68-95-99.7 rule states that approximately 68% of data in a normal distribution lies within one standard deviation of the mean. Step 4: Therefore, the percentage of students who scored between 60 and 80 is 68%. The answer is 68%.

7. A right circular cone has a height of 12 units and a base radius of 5 units. A plane parallel to the base cuts the cone, creating a smaller cone at the top and a frustum below. If the volume of the smaller top cone is exactly 1/8 of the volume of the original cone, what is the height of this smaller cone? Answer: 6 Solution: The volume V of a cone is given by the formula V = (1/3) * π * r^2 * h. The smaller cone is similar to the original cone.
   Full step-by-step solution

   Step 1: The volume V of a cone is given by the formula V = (1/3) * π * r^2 * h. Step 2: The smaller cone is similar to the original cone. For similar shapes, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions (like height). Step 3: Let the height of the smaller cone be h_s. The ratio of the heights is h_s / 12. Step 4: The volume ratio is given as 1/8. So, (h_s / 12)^3 = 1/8. Step 5: Take the cube root of both sides: h_s / 12 = ∛(1/8) = 1/2. Step 6: Solve for h_s: h_s = 12 * (1/2) = 6. The height of the smaller cone is 6 units.