Normal Distribution Estimation

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Normal Distribution Estimation · Grade 11 · Worksheet 2

1. Isabella's reaction times are normally distributed with μ=220 milliseconds and σ=25 milliseconds. Estimate the percentage of reaction times that are greater than 270 milliseconds. Answer: 2.28 Solution: Calculate the z-score for 270 milliseconds: z = (270 - 220)/25 = 50/25 = 2.00 The z-score of 2.00 means 270 is 2 standard deviations above the mean.
   Full step-by-step solution

   Step 1: Calculate the z-score for 270 milliseconds: z = (270 - 220)/25 = 50/25 = 2.00 Step 2: The z-score of 2.00 means 270 is 2 standard deviations above the mean. Step 3: Using the standard normal distribution table, the area to the left of z = 2.00 is 0.9772. Step 4: The area to the right (greater than 270) is 1 - 0.9772 = 0.0228. Step 5: Convert to a percentage: 0.0228 × 100 = 2.28%. The answer is 2.28.

2. Liam's reaction times are normally distributed with μ = 240 ms and σ = 30 ms. Estimate the percentage of reaction times greater than 270 ms. Answer: 16 Solution: Calculate the z-score for 270 ms: z = (270 - 240) / 30 = 30 / 30 = 1. The empirical rule states that approximately 68% of data lies within 1 standard deviation of the mean (between z = -1 and z = 1).
   Full step-by-step solution

   Step 1: Calculate the z-score for 270 ms: z = (270 - 240) / 30 = 30 / 30 = 1. Step 2: The empirical rule states that approximately 68% of data lies within 1 standard deviation of the mean (between z = -1 and z = 1). Step 3: This leaves 100% - 68% = 32% of data outside this range. Step 4: Since the normal distribution is symmetric, half of this 32% lies above z = 1 and half below z = -1. Step 5: Percentage above z = 1 is 32% / 2 = 16%. The answer is 16.

3. SAT scores are normally distributed with μ=1061 and σ=116. Estimate the percentage of students scoring between 945 and 1177. Answer: 68 Solution: Calculate the z-score for 945: z = (945 - 1061)/116 = -116/116 = -1 Calculate the z-score for 1177: z = (1177 - 1061)/116 = 116/116 = 1 According to the empirical rule for normal distributions, approximately 68% of data falls within 1 standard deviation of the mean (between z = -1 and z = 1).
   Full step-by-step solution

   Step 1: Calculate the z-score for 945: z = (945 - 1061)/116 = -116/116 = -1 Step 2: Calculate the z-score for 1177: z = (1177 - 1061)/116 = 116/116 = 1 Step 3: According to the empirical rule for normal distributions, approximately 68% of data falls within 1 standard deviation of the mean (between z = -1 and z = 1). Step 4: Therefore, approximately 68% of students score between 945 and 1177. The answer is 68.

4. Emma's exam scores are normally distributed with μ=85 and σ=5. What percentage of students scored between 80 and 90? Answer: 68.27 Solution: Calculate z-score for 80: z = (80 - 85)/5 = -5/5 = -1 Calculate z-score for 90: z = (90 - 85)/5 = 5/5 = 1 Look up area to the left of z = 1 in standard normal table: 0.8413 Look up area to the left of z = -1 in standard normal table: 0.1587 Subtract to find area between z = -1 and z = 1: 0.8413…
   Full step-by-step solution

   Step 1: Calculate z-score for 80: z = (80 - 85)/5 = -5/5 = -1 Step 2: Calculate z-score for 90: z = (90 - 85)/5 = 5/5 = 1 Step 3: Look up area to the left of z = 1 in standard normal table: 0.8413 Step 4: Look up area to the left of z = -1 in standard normal table: 0.1587 Step 5: Subtract to find area between z = -1 and z = 1: 0.8413 - 0.1587 = 0.6826 Step 6: Convert to percentage: 0.6826 × 100 = 68.26% The answer is 68.27% (rounded to two decimal places).

5. Liam's reaction times are normally distributed with μ=215 milliseconds and σ=25 milliseconds. Estimate the percentage of reaction times greater than 265 milliseconds. Answer: 2.28 Solution: Calculate the z-score for 265 milliseconds: z = (265 - 215) / 25 = 50 / 25 = 2.00 The area to the left of z = 2.00 in the standard normal distribution is approximately 0.9772 (from the standard normal table).
   Full step-by-step solution

   Step 1: Calculate the z-score for 265 milliseconds: z = (265 - 215) / 25 = 50 / 25 = 2.00 Step 2: The area to the left of z = 2.00 in the standard normal distribution is approximately 0.9772 (from the standard normal table). Step 3: The area to the right (greater than 265) is 1 - 0.9772 = 0.0228. Step 4: Convert to percentage: 0.0228 × 100 = 2.28%. The answer is 2.28%.

6. Charlotte's reaction times are normally distributed with μ=272 ms and σ=17 ms. Estimate the percentage of reaction times greater than 289 ms. Answer: 15.87 Solution: Calculate the z-score for 289 ms. z = (289 - 272) / 17 = 17 / 17 = 1.00 The area to the left of z = 1.00 in the standard normal distribution is approximately 0.8413 (from standard normal table).
   Full step-by-step solution

   Step 1: Calculate the z-score for 289 ms. z = (289 - 272) / 17 = 17 / 17 = 1.00 Step 2: The area to the left of z = 1.00 in the standard normal distribution is approximately 0.8413 (from standard normal table). Step 3: The area to the right (greater than 289 ms) is 1 - 0.8413 = 0.1587. Step 4: Convert to a percentage: 0.1587 × 100 = 15.87%. The answer is 15.87.

7. A Ferris wheel with a diameter of 40 meters completes one full rotation every 2 minutes. A passenger boards at the lowest point, which is 5 meters above ground level. The height of the passenger above ground can be modeled by a sinusoidal function h(t) = a + b·sin(c·t + d), where t is time in minutes after boarding. Determine the exact values of parameters a, b, c, and d for this height function. Answer: a = 25, b = 20, c = π, d = -π/2 Solution: Diameter = 40 m → radius = 20 m. Lowest point = 5 m above ground. Highest point = 5 m + 40 m = 45 m above ground.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understanding the problem** Diameter = 40 m → radius = 20 m. Lowest point = 5 m above ground. Highest point = 5 m + 40 m = 45 m above ground. Period = 2 minutes. Model: h(t) = a + b·sin(c·t + d) --- **Step 2: Determine a and b** For a sine function of the form a + b·sin(θ), - a = vertical shift (midline height) - b = amplitude (distance from midline to max or min) Midline height = (lowest + highest)/2 = (5 + 45)/2 = 50/2 = 25 m. So a = 25. Amplitude = (highest - lowest)/2 = (45 - 5)/2 = 40/2 = 20 m. So b = 20. --- **Step 3: Determine c** Period T = 2 minutes. For sine function sin(c·t + d), period = 2π / c. So 2π / c = 2 → c = π. --- **Step 4: Determine d** We know: h(t) = 25 + 20·sin(π·t + d) At t = 0, passenger is at lowest point = 5 m above ground. So h(0) = 5. Plug in: 5 = 25 + 20·sin(π·0 + d) 5 = 25 + 20·sin(d) 20·sin(d) = 5 - 25 = -20 sin(d) = -1 --- **Step 5: Solve for d** sin(d) = -1 → d = -π/2 + 2πn. We can take the principal value d = -π/2. Check: h(t) = 25 + 20·sin(π·t - π/2) At t=0: sin(-π/2) = -1 → h(0) = 25 - 20 = 5 (correct, lowest point). At t=0.5: sin(π/2 - π/2) = sin(0) = 0 → h=25 (midline, rising). At t=1: sin(π - π/2) = sin(π/2) = 1 → h=45 (highest point). Works fine. --- **Final answer:** a = 25, b = 20, c = π, d = -π/2

8. A pharmaceutical company is testing a new medication and wants to estimate the proportion of patients who experience significant improvement. In a clinical trial with 320 randomly selected patients, 224 showed significant improvement. Construct a 95% confidence interval for the true population proportion of patients who would experience significant improvement with this medication. (Use z* = 1.96 for the critical value) Answer: 0.644 to 0.756 Solution: Confidence intervals for population proportions use the sample proportion as a point estimate and then add/subtract a margin of error that depends on the desired confidence level and sample size.
   Full step-by-step solution

   Confidence intervals for population proportions use the sample proportion as a point estimate and then add/subtract a margin of error that depends on the desired confidence level and sample size. The margin of error accounts for sampling variability and decreases with larger sample sizes, providing more precise estimates.