Normal Distribution

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Answer Key & Explanations

Normal Distribution · Grade 11 · Worksheet 2

1. A pharmaceutical company tests a new medication and finds the recovery times for patients follow a normal distribution with a mean of 14 days and a standard deviation of 2.5 days. Dr. Chen is treating a patient who recovered in 19 days. What percentage of patients would be expected to recover in more time than Dr. Chen's patient? Answer: 2.28% Solution: We have a normal distribution of recovery times with: Mean (μ) = 14 days Standard deviation (σ) = 2.5 days Dr. Chen's patient recovered in 19 days. We want the percentage of patients who recover in *more* than 19 days.
   Full step-by-step solution

   Step 1: Understand the problem We have a normal distribution of recovery times with: Mean (μ) = 14 days Standard deviation (σ) = 2.5 days Dr. Chen's patient recovered in 19 days. We want the percentage of patients who recover in *more* than 19 days. --- Step 2: Calculate the z-score The z-score tells us how many standard deviations above the mean 19 days is. Formula: z = (x - μ) / σ Here: x = 19 μ = 14 σ = 2.5 z = (19 - 14) / 2.5 z = 5 / 2.5 z = 2 So 19 days is 2 standard deviations above the mean. --- Step 3: Interpret the z-score We want the probability of recovery time > 19 days, which is P(Z > 2) in the standard normal distribution. From standard normal distribution tables (or the empirical rule): - About 95% of data lies within 2 standard deviations of the mean (between z = -2 and z = 2). - That means about 5% lies outside this range (2.5% in each tail). So P(Z > 2) ≈ 0.0228 or 2.28%. --- Step 4: Verify with more precise reasoning Using a z-table: P(Z < 2) = 0.9772 Therefore, P(Z > 2) = 1 - 0.9772 = 0.0228 Convert to percentage: 0.0228 × 100 = 2.28% --- Step 5: Final answer The percentage of patients expected to recover in more time than Dr. Chen's patient is 2.28%.

2. Normal: μ=72, σ=9. What % between 54 and 90? Answer: 95 Solution: Calculate the z-score for 54: (54 - 72)/9 = -18/9 = -2. Step 2: Calculate the z-score for 90: (90 - 72)/9 = 18/9 = 2.
   Full step-by-step solution

   Step 1: Calculate the z-score for 54: (54 - 72)/9 = -18/9 = -2. Step 2: Calculate the z-score for 90: (90 - 72)/9 = 18/9 = 2. Step 3: According to the 68-95-99.7 rule, approximately 95% of data falls within 2 standard deviations of the mean. Step 4: Therefore, approximately 95% of values lie between 54 and 90. The answer is 95.

3. Noah is analyzing the weights of adult male African elephants in a reserve. The weights follow a normal distribution with a mean of 5000 kg and a standard deviation of 400 kg. Using the 68-95-99.7 rule, what percentage of these elephants weigh between 4200 kg and 5800 kg? Answer: 95% Solution: Identify the mean (μ = 5000 kg) and standard deviation (σ = 400 kg). Find how many standard deviations 4200 kg is from the mean: (5000 - 4200) / 400 = 800 / 400 = 2 standard deviations below the mean.
   Full step-by-step solution

   Step 1: Identify the mean (μ = 5000 kg) and standard deviation (σ = 400 kg). Step 2: Find how many standard deviations 4200 kg is from the mean: (5000 - 4200) / 400 = 800 / 400 = 2 standard deviations below the mean. Step 3: Find how many standard deviations 5800 kg is from the mean: (5800 - 5000) / 400 = 800 / 400 = 2 standard deviations above the mean. Step 4: The 68-95-99.7 rule states that approximately 95% of data in a normal distribution lies within 2 standard deviations of the mean. Step 5: Therefore, about 95% of the elephants weigh between 4200 kg and 5800 kg. Answer: 95%

4. Emma is analyzing the distribution of scores on a Grade 11 mathematics exam. The scores are normally distributed with a mean of 69 and a standard deviation of 7. According to the 68-95-99.7 rule, approximately what percentage of students scored between 55 and 76? Answer: 81.5 Solution: Find the z-scores for the boundaries. The mean is 69 and the standard deviation is 7. For 55: z = (55 - 69) / 7 = -14 / 7 = -2.
   Full step-by-step solution

   Step 1: Find the z-scores for the boundaries. The mean is 69 and the standard deviation is 7. For 55: z = (55 - 69) / 7 = -14 / 7 = -2. For 76: z = (76 - 69) / 7 = 7 / 7 = 1. Step 2: Interpret the z-scores using the 68-95-99.7 rule. - Between z = -2 and z = 2 is 95% of the data. So, between z = -2 and z = 0 is half of that, which is 47.5%. - Between z = -1 and z = 1 is 68% of the data. So, between z = 0 and z = 1 is half of that, which is 34%. Step 3: Add the percentages for the range from z = -2 to z = 1. Percentage = (percentage from z = -2 to 0) + (percentage from z = 0 to 1) = 47.5% + 34% = 81.5%. The answer is 81.5.

5. 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, passing through all three vertices. What is the area of this circumscribed circle? (Use π = 3.14) Answer: 78.5 Solution: The triangle has vertices at (0,0), (6,0), and (6,8). Points (0,0) and (6,0) are on the x-axis, so the side between them is horizontal with length 6.
   Full step-by-step solution

   Step 1: Understand the triangle and circle The triangle has vertices at (0,0), (6,0), and (6,8). Points (0,0) and (6,0) are on the x-axis, so the side between them is horizontal with length 6. Points (6,0) and (6,8) are on a vertical line, so the side between them is vertical with length 8. The third side from (0,0) to (6,8) is the hypotenuse. Step 2: Verify it is a right triangle The sides: Horizontal leg length = 6 Vertical leg length = 8 Slopes: from (0,0) to (6,0) is 0, from (6,0) to (6,8) is undefined (vertical). So the right angle is at (6,0). Thus the hypotenuse is from (0,0) to (6,8). Step 3: Find the hypotenuse length Distance between (0,0) and (6,8): sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10. So the sides are 6, 8, 10. Step 4: Circle circumscribed around a right triangle A key fact: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. The center of the circle is the midpoint of the hypotenuse, and the radius is half the hypotenuse. Step 5: Find the radius Hypotenuse = 10, so diameter = 10, radius r = 10/2 = 5. Step 6: Area of the circle Area = π * r^2 = 3.14 * (5^2) = 3.14 * 25. Step 7: Calculate 3.14 * 25 = 78.5. Final answer: 78.5

6. Mere is analyzing the weights of adult male kakapo in a conservation sanctuary. The weights follow a normal distribution with a mean of 2.2 kg and a standard deviation of 0.3 kg. The sanctuary's veterinary team wants to identify birds that are in the top 0.15% of the weight distribution for a special health study. Using the 68-95-99.7 rule, what is the minimum weight a kakapo must have to be included in this study? Answer: 3.1 kg Solution: Identify the mean and standard deviation. Mean (μ) = 2.2 kg Standard deviation (σ) = 0.3 kg Understand the 68-95-99.7 rule.
   Full step-by-step solution

   Step 1: Identify the mean and standard deviation. Mean (μ) = 2.2 kg Standard deviation (σ) = 0.3 kg Step 2: Understand the 68-95-99.7 rule. Within 1 standard deviation: 68% of data Within 2 standard deviations: 95% of data Within 3 standard deviations: 99.7% of data The total area under the normal curve is 100%. The area beyond 3 standard deviations is 100% - 99.7% = 0.3%. Since the normal distribution is symmetric, this 0.3% is split equally into two tails: - Lower tail (below 3 standard deviations below the mean): 0.15% - Upper tail (above 3 standard deviations above the mean): 0.15% Step 3: Find the weight that is 3 standard deviations above the mean. 3 standard deviations above the mean = μ + 3σ = 2.2 + 3(0.3) = 2.2 + 0.9 = 3.1 kg Step 4: Interpret the result. The top 0.15% of kakapo have weights greater than 3 standard deviations above the mean. Therefore, any kakapo weighing more than 3.1 kg is in the top 0.15%. The minimum weight for inclusion in the study is 3.1 kg. The answer is 3.1 kg.

7. Noah is analyzing a set of test scores that follow a normal distribution with a mean of 76 and a standard deviation of 6. A histogram of the scores is shaped like a symmetric bell curve centered at 76. What percentage of the test scores lie between 64 and 88? Answer: 95 Solution: The mean is μ = 76 and the standard deviation is σ = 6. The lower bound 64 is (76 - 64) = 12 units below the mean. Since 12 = 2 × 6, this is 2 standard deviations below the mean.
   Full step-by-step solution

   Step 1: The mean is μ = 76 and the standard deviation is σ = 6. Step 2: The lower bound 64 is (76 - 64) = 12 units below the mean. Since 12 = 2 × 6, this is 2 standard deviations below the mean. Step 3: The upper bound 88 is (88 - 76) = 12 units above the mean. Since 12 = 2 × 6, this is 2 standard deviations above the mean. Step 4: According to the 68-95-99.7 rule (empirical rule) for normal distributions, approximately 95% of the data falls within 2 standard deviations of the mean. Step 5: Therefore, the percentage of test scores between 64 and 88 is 95%.