Normal Distribution Estimation

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Normal Distribution Estimation · Grade 11 · Worksheet 3

1. Sophia's reaction times are normally distributed with μ=220 milliseconds and σ=25 milliseconds. Estimate the percentage of reaction times greater than 270 milliseconds. Answer: 2.28 Solution: Calculate the z-score for x = 270: z = (270 - 220)/25 = 50/25 = 2.0 The area to the left of z = 2.0 in the standard normal distribution is approximately 0.9772 (from standard normal table).
   Full step-by-step solution

   Step 1: Calculate the z-score for x = 270: z = (270 - 220)/25 = 50/25 = 2.0 Step 2: The area to the left of z = 2.0 in the standard normal distribution is approximately 0.9772 (from standard normal table). Step 3: The area to the right (greater than 270) is 1 - 0.9772 = 0.0228. Step 4: Convert to percentage: 0.0228 × 100 = 2.28%. The answer is 2.28%.

2. Isabella's test scores are normally distributed with μ=88 and σ=12. What percentage of students scored between 76 and 100? Answer: 68.26 Solution: Calculate z-score for 76: z = (76 - 88)/12 = -12/12 = -1.00 Calculate z-score for 100: z = (100 - 88)/12 = 12/12 = 1.00 Look up P(z < 1.00) in standard normal table: 0.8413 Look up P(z < -1.00) in standard normal table: 0.1587 Calculate percentage between z = -1.00 and z = 1.00: 0.8413 - 0.1587…
   Full step-by-step solution

   Step 1: Calculate z-score for 76: z = (76 - 88)/12 = -12/12 = -1.00 Step 2: Calculate z-score for 100: z = (100 - 88)/12 = 12/12 = 1.00 Step 3: Look up P(z < 1.00) in standard normal table: 0.8413 Step 4: Look up P(z < -1.00) in standard normal table: 0.1587 Step 5: Calculate percentage between z = -1.00 and z = 1.00: 0.8413 - 0.1587 = 0.6826 Step 6: Convert to percentage: 0.6826 × 100 = 68.26% The answer is 68.26.

3. A city is planning a new public transit system and needs to estimate what percentage of residents would use it regularly. They survey 400 randomly selected residents and find that 68% express interest. Construct a 95% confidence interval for the true population proportion of residents who would use the transit system regularly. (Use z* = 1.96 for the critical value) Answer: (0.634, 0.726) Solution: Identify the sample proportion. We are told that 68% of the 400 surveyed residents express interest. Find the standard error of the proportion.
   Full step-by-step solution

   Step 1: Identify the sample proportion. We are told that 68% of the 400 surveyed residents express interest. So, the sample proportion p-hat = 0.68. Step 2: Find the standard error of the proportion. The formula for the standard error (SE) is: SE = sqrt( (p-hat * (1 - p-hat)) / n ) Substitute the values: SE = sqrt( (0.68 * (1 - 0.68)) / 400 ) First, 1 - 0.68 = 0.32. Then, 0.68 * 0.32 = 0.2176. Then, 0.2176 / 400 = 0.000544. Now, SE = sqrt(0.000544) ≈ 0.023323. Step 3: Determine the margin of error (ME). ME = z* × SE Given z* = 1.96 for 95% confidence: ME = 1.96 × 0.023323 ≈ 0.045713. Step 4: Construct the confidence interval. The formula is: p-hat ± ME Lower bound = 0.68 - 0.045713 ≈ 0.634287 Upper bound = 0.68 + 0.045713 ≈ 0.725713 Step 5: Round to three decimal places as typical for such intervals. Lower bound ≈ 0.634 Upper bound ≈ 0.726 Final 95% confidence interval: (0.634, 0.726) This means we are 95% confident that the true proportion of all residents who would use the transit system regularly is between 63.4% and 72.6%.

4. A city's population growth follows the logistic model P(t) = 12000 / (1 + 3e^(-0.08t)), where t is time in years since 2020. What is the carrying capacity of the environment for this population? Answer: 12000 Solution: The standard form of the logistic growth model is: P(t) = L / (1 + C * e^(-k t)) where L is the carrying capacity, C is a constant related to initial conditions, k is the growth rate, and t is time.
   Full step-by-step solution

   Step 1: Understand the logistic growth model The standard form of the logistic growth model is: P(t) = L / (1 + C * e^(-k t)) where L is the carrying capacity, C is a constant related to initial conditions, k is the growth rate, and t is time. Step 2: Identify the given function The problem gives: P(t) = 12000 / (1 + 3 * e^(-0.08 t)) Step 3: Compare with the standard form Comparing P(t) = 12000 / (1 + 3 * e^(-0.08 t)) with P(t) = L / (1 + C * e^(-k t)), we see: - L = 12000 - C = 3 - k = 0.08 Step 4: Interpret the meaning of L In the logistic model, L is the carrying capacity — the maximum population the environment can sustain in the long run. Step 5: Verify by considering the limit as t → ∞ As t → ∞, e^(-0.08 t) → 0, so: P(t) → 12000 / (1 + 0) = 12000 This confirms that the population approaches 12000 in the long term. Step 6: Conclusion The carrying capacity is 12000.

5. A right circular cone has a height of 12 cm and a base radius of 5 cm. A plane cuts the cone parallel to its base, creating a smaller cone at the top with height 4 cm. What is the volume of the frustum (the remaining portion between the two parallel planes)? Answer: 700π/3 cm³ Solution: When a cone is cut by a plane parallel to its base, the resulting cross-section is similar to the original base. This similarity relationship allows us to determine the dimensions of the smaller cone that was removed.
   Full step-by-step solution

   When a cone is cut by a plane parallel to its base, the resulting cross-section is similar to the original base. This similarity relationship allows us to determine the dimensions of the smaller cone that was removed. The volume of any cone is proportional to the cube of its linear dimensions when comparing similar cones. For frustum problems, the most straightforward approach is to calculate the volume of the complete cone and subtract the volume of the removed cone portion.

6. Matiu's reaction times are normally distributed with μ=240 milliseconds and σ=30 milliseconds. Estimate the percentage of reaction times between 210 and 270 milliseconds. Answer: 68 Solution: Calculate the z-score for 210: z = (210 - 240)/30 = -30/30 = -1. Calculate the z-score for 270: z = (270 - 240)/30 = 30/30 = 1. The interval from z = -1 to z = 1 represents data within 1 standard deviation of the mean.
   Full step-by-step solution

   Step 1: Calculate the z-score for 210: z = (210 - 240)/30 = -30/30 = -1. Step 2: Calculate the z-score for 270: z = (270 - 240)/30 = 30/30 = 1. Step 3: The interval from z = -1 to z = 1 represents data within 1 standard deviation of the mean. Step 4: According to the empirical rule, approximately 68% of data in a normal distribution lies within 1 standard deviation of the mean. Step 5: Therefore, about 68% of reaction times are between 210 and 270 milliseconds. The answer is 68.

7. A Ferris wheel with a diameter of 50 meters completes one full rotation every 3 minutes. The boarding platform is 3 meters above ground level, and a passenger boards at the lowest point. The height of a 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=28,b=25,c=2π/3,d=-π/2 Solution: The Ferris wheel has diameter 50 m, so radius = 25 m The center of the wheel is at height = platform height + radius = 3 + 25 = 28 m Therefore, a = 28 Therefore, b = 25 The wheel completes one full rotation every 3 minutes Angular frequency c = 2π / period = 2π / 3 The passenger boards at the…
   Full step-by-step solution

   Step 1: Determine parameter a (vertical shift) The Ferris wheel has diameter 50 m, so radius = 25 m The center of the wheel is at height = platform height + radius = 3 + 25 = 28 m Therefore, a = 28 Step 2: Determine parameter b (amplitude) The amplitude equals the radius of the wheel Therefore, b = 25 Step 3: Determine parameter c (angular frequency) The wheel completes one full rotation every 3 minutes Angular frequency c = 2π / period = 2π / 3 Step 4: Determine parameter d (phase shift) The passenger boards at the lowest point, which corresponds to -π/2 in the sine function Therefore, d = -π/2 Step 5: Final parameter values a = 28, b = 25, c = 2π/3, d = -π/2 The complete height function is h(t) = 28 + 25·sin((2π/3)·t - π/2)

8. Olivia's test 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 Using the standard normal distribution table, the area to the left of z = 1 is 0.8413 The area to the left of z = -1 is 0.1587 The area between z = -1 and z = 1 is 0.8413 - 0.1587 = 0.6826…
   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: Using the standard normal distribution table, the area to the left of z = 1 is 0.8413 Step 4: The area to the left of z = -1 is 0.1587 Step 5: The area between z = -1 and z = 1 is 0.8413 - 0.1587 = 0.6826 Step 6: Convert to percentage: 0.6826 × 100 = 68.26% Step 7: Using the empirical rule, approximately 68.27% of data falls within 1 standard deviation of the mean Final answer: 68.27%