Normal Distribution

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Answer Key & Explanations

Normal Distribution · Grade 11 · Worksheet 1

1. Hana is analyzing the distribution of the weights of a rare species of fish in a lake. The weights are normally distributed with a mean of 28 kg and a standard deviation of 5 kg. A normal distribution curve is drawn with the mean at 28 kg. Shade the region on the curve that represents fish weighing between 18 kg and 38 kg. Using the 68-95-99.7 rule, approximately what percentage of the fish fall in this shaded region? Answer: 95 Solution: The mean is 28 kg and the standard deviation is 5 kg. Calculate the z-scores: for 18 kg, z = (18 - 28)/5 = -10/5 = -2 (2 standard deviations below the mean).
   Full step-by-step solution

   Step 1: The mean is 28 kg and the standard deviation is 5 kg. Calculate the z-scores: for 18 kg, z = (18 - 28)/5 = -10/5 = -2 (2 standard deviations below the mean). For 38 kg, z = (38 - 28)/5 = 10/5 = 2 (2 standard deviations above the mean). Step 2: According to the 68-95-99.7 rule, 95% of the data falls within 2 standard deviations of the mean (from z = -2 to z = 2). Step 3: The region from 18 kg to 38 kg is exactly from z = -2 to z = 2, so the percentage is 95%. The answer is 95.

2. 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%.

3. 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.

4. Olivia is studying the distribution of the weights of adult male koalas in a wildlife sanctuary. The weights are normally distributed with a mean of 11 kg and a standard deviation of 3 kg. A normal distribution curve is drawn with the mean marked at 11 kg. On this curve, shade the region that represents koalas weighing between 5 kg and 17 kg. Using the 68-95-99.7 rule, approximately what percentage of all koalas fall in this shaded region? Answer: 95 Solution: The mean (μ) is 11 kg and the standard deviation (σ) is 3 kg. Step 2: Calculate the lower boundary's distance from the mean: 11 - 5 = 6 kg. Since 6 / 3 = 2, the lower boundary is 2 standard deviations below the mean.
   Full step-by-step solution

   Step 1: The mean (μ) is 11 kg and the standard deviation (σ) is 3 kg. Step 2: Calculate the lower boundary's distance from the mean: 11 - 5 = 6 kg. Since 6 / 3 = 2, the lower boundary is 2 standard deviations below the mean. Step 3: Calculate the upper boundary's distance from the mean: 17 - 11 = 6 kg. Since 6 / 3 = 2, the upper boundary is 2 standard deviations above the mean. Step 4: The region from 5 kg to 17 kg represents all data within 2 standard deviations of the mean (μ ± 2σ). Step 5: According to the 68-95-99.7 rule for normal distributions, approximately 95% of the data falls within 2 standard deviations of the mean. Therefore, the percentage of koalas weighing between 5 kg and 17 kg is approximately 95%. The answer is 95.

5. Mason is analyzing the battery life of a new laptop model. The battery life (in hours) follows a normal distribution with a mean of 7.2 hours and a standard deviation of 0.7 hours. The manufacturer wants to guarantee that the battery will last at least 5.8 hours for all laptops sold. Using the 68-95-99.7 rule, what percentage of laptops are expected to have a battery life below this guaranteed minimum of 5.8 hours? Answer: 2.5% Solution: Identify the mean (μ = 7.2 hours) and standard deviation (σ = 0.7 hours). Find how many standard deviations 5.8 hours is from the mean: difference = 7.2 - 5.8 = 1.4 hours. Number of standard deviations = 1.4 / 0.7 = 2.
   Full step-by-step solution

   Step 1: Identify the mean (μ = 7.2 hours) and standard deviation (σ = 0.7 hours). Step 2: Find how many standard deviations 5.8 hours is from the mean: difference = 7.2 - 5.8 = 1.4 hours. Number of standard deviations = 1.4 / 0.7 = 2. So, 5.8 hours is 2 standard deviations below the mean. Step 3: According to the 68-95-99.7 rule, approximately 95% of data falls within 2 standard deviations of the mean (between 5.8 and 8.6 hours). Step 4: The remaining 5% of data (100% - 95%) is split equally between the two tails (below 5.8 hours and above 8.6 hours). Step 5: The percentage below 5.8 hours is 5% / 2 = 2.5%. Therefore, 2.5% of laptops are expected to have a battery life below 5.8 hours. The answer is 2.5%.

6. √(25) + log₂(8) = ? Answer: 8 Solution: Calculate the square root: √(25) = 5 Calculate the logarithm: log₂(8) means 2 raised to what power equals 8? Since 2³ = 8, log₂(8) = 3 Add the results: 5 + 3 = 8 The answer is 8.
   Full step-by-step solution

   Step 1: Calculate the square root: √(25) = 5 Step 2: Calculate the logarithm: log₂(8) means 2 raised to what power equals 8? Since 2³ = 8, log₂(8) = 3 Step 3: Add the results: 5 + 3 = 8 The answer is 8.