Normal Distribution

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Answer Key & Explanations

Normal Distribution · Grade 11 · Worksheet 2

1. Sophia is analyzing the scores from a national mathematics competition. The scores are normally distributed with a mean of 61 points and a standard deviation of 6 points. The top 16% of students receive a certificate of excellence. Using the 68-95-99.7 rule, determine the minimum score a student must achieve to earn the certificate. Answer: 67 points Solution: The scores are normally distributed with mean μ = 61 and standard deviation σ = 6. The top 16% means the cutoff is at the 84th percentile (since 100% - 16% = 84% of scores are below this point).
   Full step-by-step solution

   Step 1: The scores are normally distributed with mean μ = 61 and standard deviation σ = 6. Step 2: The top 16% means the cutoff is at the 84th percentile (since 100% - 16% = 84% of scores are below this point). Step 3: Using the 68-95-99.7 rule: 68% of data lies within 1 standard deviation of the mean. This means 32% lies outside that range, with 16% in each tail (since the normal distribution is symmetric). Step 4: Therefore, the top 16% begins exactly 1 standard deviation above the mean. Step 5: Calculate the cutoff score: μ + 1σ = 61 + 6 = 67. The minimum score needed for the certificate is 67 points.

2. 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%.

3. A triangular prism has a right triangle base with legs measuring 6 cm and 8 cm, and a height of 15 cm. The prism is positioned so that the triangular bases are vertical. What is the total surface area of this prism? Answer: 408 cm² Solution: We have a triangular prism with a right triangle base. Legs of the base triangle: 6 cm and 8 cm. Height of prism (length between triangular bases): 15 cm.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the shape** We have a triangular prism with a right triangle base. Legs of the base triangle: 6 cm and 8 cm. Height of prism (length between triangular bases): 15 cm. Triangular bases are vertical — this means the lateral faces are rectangles whose heights are 15 cm. --- **Step 2: Find the hypotenuse of the triangular base** Right triangle legs: 6 cm and 8 cm. Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm. --- **Step 3: Find the area of one triangular base** Area of triangle = (1/2) × base × height (legs are perpendicular) = (1/2) × 6 × 8 = 24 cm². There are 2 triangular bases: total base area = 2 × 24 = 48 cm². --- **Step 4: Find the lateral surface area** Lateral surface area = perimeter of base triangle × height of prism. Perimeter of base triangle = 6 + 8 + 10 = 24 cm. Lateral surface area = 24 × 15 = 360 cm². --- **Step 5: Total surface area** Total surface area = lateral surface area + area of two triangular bases = 360 + 48 = 408 cm². --- **Final answer:** 408 cm²

4. Sophia is analyzing the battery life of a new smartphone model. The battery life (in hours of continuous video playback) follows a normal distribution with a mean of 16 hours and a standard deviation of 1 hour. Using the 68-95-99.7 rule, what percentage of smartphones have a battery life between 14 hours and 17 hours? Answer: 81.5% Solution: Identify the mean and standard deviation. Mean = 16 hours, standard deviation = 1 hour. Step 2: Determine the z-scores.
   Full step-by-step solution

   Step 1: Identify the mean and standard deviation. Mean = 16 hours, standard deviation = 1 hour. Step 2: Determine the z-scores. 14 hours is 2 standard deviations below the mean (z = -2). 17 hours is 1 standard deviation above the mean (z = +1). Step 3: Apply the 68-95-99.7 rule. Approximately 95% of data lies within 2 standard deviations of the mean (between 14 and 18 hours). Approximately 68% lies within 1 standard deviation (between 15 and 17 hours). Step 4: Find the percentage between 14 and 17. From 14 to the mean (16) is half of the 95% range, which is 47.5%. From the mean to 17 is half of the 68% range, which is 34%. Step 5: Add: 47.5% + 34% = 81.5%. The answer is 81.5%.

5. Aroha is a marine biologist studying the weights of adult snapper in a large marine reserve. The weights follow a normal distribution with a mean of 2.8 kg and a standard deviation of 0.6 kg. Marine park regulations state that any snapper weighing less than 2.2 kg or more than 3.4 kg must be released immediately if caught. Using the 68-95-99.7 rule, estimate the percentage of adult snapper in the reserve that would be kept (i.e., NOT released) by a fisher who follows these regulations. Answer: 68% Solution: Identify the mean (μ = 2.8 kg) and standard deviation (σ = 0.6 kg). Find the z-scores for the cutoff weights.
   Full step-by-step solution

   Step 1: Identify the mean (μ = 2.8 kg) and standard deviation (σ = 0.6 kg). Step 2: Find the z-scores for the cutoff weights. For 2.2 kg: z = (2.2 - 2.8) / 0.6 = -0.6 / 0.6 = -1 For 3.4 kg: z = (3.4 - 2.8) / 0.6 = 0.6 / 0.6 = 1 Step 3: The cutoff weights are exactly 1 standard deviation below and above the mean. Step 4: Apply the 68-95-99.7 rule. Approximately 68% of data in a normal distribution lies within 1 standard deviation of the mean (between μ - σ and μ + σ). Step 5: Therefore, the percentage of snapper that weigh between 2.2 kg and 3.4 kg is approximately 68%. Step 6: Since fishers keep snapper within this range, 68% of adult snapper would be kept. Answer: 68%

6. 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.

7. The heights of sunflowers in a field follow a normal distribution with a mean of 185 cm and a standard deviation of 12 cm. What percentage of sunflowers have heights between 161 cm and 209 cm? Answer: 95 Solution: Calculate the z-score for 161 cm: z = (161 - 185) / 12 = -24 / 12 = -2 Calculate the z-score for 209 cm: z = (209 - 185) / 12 = 24 / 12 = 2 According to the empirical rule for normal distributions, approximately 95% of data falls within 2 standard deviations of the mean (between z = -2 and z = 2).
   Full step-by-step solution

   Step 1: Calculate the z-score for 161 cm: z = (161 - 185) / 12 = -24 / 12 = -2 Step 2: Calculate the z-score for 209 cm: z = (209 - 185) / 12 = 24 / 12 = 2 Step 3: According to the empirical rule for normal distributions, approximately 95% of data falls within 2 standard deviations of the mean (between z = -2 and z = 2). Step 4: Therefore, approximately 95% of sunflowers have heights between 161 cm and 209 cm. The answer is 95.