Normal Distribution Estimation

lessonbunny.com

Answer Key & Explanations

Normal Distribution Estimation · Grade 11 · Worksheet 3

1. Hana's reaction times are normally distributed with μ=240 ms and σ=32 ms. Estimate the percentage of reaction times between 208 ms and 272 ms. Answer: 68 Solution: Calculate the z-score for 208 ms: z = (208 - 240)/32 = -32/32 = -1. Step 2: Calculate the z-score for 272 ms: z = (272 - 240)/32 = 32/32 = 1.
   Full step-by-step solution

   Step 1: Calculate the z-score for 208 ms: z = (208 - 240)/32 = -32/32 = -1. Step 2: Calculate the z-score for 272 ms: z = (272 - 240)/32 = 32/32 = 1. Step 3: According to the empirical rule, approximately 68% of data in a normal distribution lies within 1 standard deviation of the mean (between z = -1 and z = 1). Step 4: Therefore, approximately 68% of Hana's reaction times are between 208 ms and 272 ms. The answer is 68.

2. 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.

3. Matiu is a botanist studying the heights of a particular species of tree in a large forest. The heights are normally distributed with a mean of 14.2 meters and a standard deviation of 2.6 meters. Estimate the percentage of trees in the forest that are between 11.6 meters and 16.8 meters tall. Answer: 68.26% Solution: Identify the mean (mu = 14.2 m) and standard deviation (sigma = 2.6 m). Calculate the z-score for the lower bound (11.6 m): z = (11.6 - 14.2) / 2.6 = (-2.6) / 2.6 = -1.
   Full step-by-step solution

   Step 1: Identify the mean (mu = 14.2 m) and standard deviation (sigma = 2.6 m). Step 2: Calculate the z-score for the lower bound (11.6 m): z = (11.6 - 14.2) / 2.6 = (-2.6) / 2.6 = -1. Step 3: Calculate the z-score for the upper bound (16.8 m): z = (16.8 - 14.2) / 2.6 = 2.6 / 2.6 = 1. Step 4: The interval from z = -1 to z = 1 represents one standard deviation from the mean on either side. Step 5: According to the empirical rule for normal distributions, approximately 68.26% of the data lies within one standard deviation of the mean. Step 6: Therefore, the estimated percentage of trees with heights between 11.6 m and 16.8 m is 68.26%. The answer is 68.26%.

4. Kaia's reaction times are normally distributed with μ = 265 milliseconds and σ = 15 milliseconds. Estimate the percentage of reaction times greater than 295 milliseconds. Answer: 2.28 Solution: Calculate the z-score for 295 milliseconds: z = (295 - 265) / 15 = 30 / 15 = 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 295 milliseconds: z = (295 - 265) / 15 = 30 / 15 = 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 295) is 1 - 0.9772 = 0.0228. Step 4: Convert to percentage: 0.0228 × 100 = 2.28%. The answer is 2.28.

5. Matiu, a quality control engineer at a manufacturing plant, measures the tensile strength (in MPa) of a batch of steel rods produced by a new process. He finds that the tensile strengths are normally distributed with a mean of 450 MPa and a standard deviation of 24 MPa. The company's specification requires that the rods have a tensile strength between 426 MPa and 498 MPa to be considered acceptable. Estimate the percentage of rods from this production process that meet the specification. Answer: 81.85% Solution: Calculate the z-score for the lower bound (426 MPa). z1 = (426 - 450) / 24 = -24 / 24 = -1.00 Calculate the z-score for the upper bound (498 MPa).
   Full step-by-step solution

   Step 1: Calculate the z-score for the lower bound (426 MPa). z1 = (426 - 450) / 24 = -24 / 24 = -1.00 Step 2: Calculate the z-score for the upper bound (498 MPa). z2 = (498 - 450) / 24 = 48 / 24 = 2.00 Step 3: Find the area to the left of z = -1.00 using a z-table. This area is 0.1587. Step 4: Find the area to the left of z = 2.00 using a z-table. This area is 0.9772. Step 5: Subtract the smaller area from the larger area to find the area between the two z-scores. Area between = 0.9772 - 0.1587 = 0.8185 Step 6: Convert this proportion to a percentage. Percentage = 0.8185 * 100% = 81.85% Therefore, approximately 81.85% of the steel rods are estimated to meet the tensile strength specification.

6. 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%.

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)