Normal Distribution Estimation

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Answer Key & Explanations

Normal Distribution Estimation · Grade 11 · Worksheet 1

1. A city planner is analyzing traffic flow data and finds that the number of cars passing through an intersection follows a sinusoidal pattern throughout the day. The function modeling this is f(t) = 250sin(π/12(t - 6)) + 350, where t is the time in hours after midnight. During what time intervals is the traffic flow above 500 cars per hour? Answer: 10 < t < 14 Solution: Sinusoidal functions model periodic phenomena like traffic patterns. The general form f(t) = Asin(B(t - C)) + D has amplitude A, period determined by B, horizontal shift C, and vertical shift D.
   Full step-by-step solution

   Sinusoidal functions model periodic phenomena like traffic patterns. The general form f(t) = Asin(B(t - C)) + D has amplitude A, period determined by B, horizontal shift C, and vertical shift D. To find when the function exceeds a certain value, you solve an inequality involving the trigonometric function, considering the periodic nature of the solution.

2. A pharmaceutical company is testing a new medication and wants to estimate what percentage of patients will experience a specific side effect. In a clinical trial with 320 randomly selected patients, 24% reported experiencing the side effect. Construct a 90% confidence interval for the true population proportion of patients who would experience this side effect. (Use z* = 1.645 for the critical value) Answer: 0.200 to 0.280 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 sample size and desired confidence level.
   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 sample size and desired confidence level. The margin of error decreases with larger sample sizes and increases with higher confidence levels, reflecting the trade-off between precision and certainty in statistical estimation.

3. Charlotte's reaction times are normally distributed with μ=272 ms and σ=17 ms. Estimate the percentage of reaction times between 238 ms and 306 ms. Answer: 95 Solution: Calculate the z-score for 238 ms: z = (238 - 272)/17 = -34/17 = -2. Step 2: Calculate the z-score for 306 ms: z = (306 - 272)/17 = 34/17 = 2.
   Full step-by-step solution

   Step 1: Calculate the z-score for 238 ms: z = (238 - 272)/17 = -34/17 = -2. Step 2: Calculate the z-score for 306 ms: z = (306 - 272)/17 = 34/17 = 2. Step 3: According to the empirical rule, approximately 95% of data in a normal distribution lies within 2 standard deviations of the mean (between z = -2 and z = 2). Step 4: Therefore, approximately 95% of Charlotte's reaction times are between 238 ms and 306 ms. The answer is 95.

4. A right circular cone has a height of 12 cm and a base radius of 5 cm. A plane parallel to the base cuts the cone, 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 smaller cone is similar to the original cone. This similarity allows us to find the radius of the smaller cone using proportional relationships. The formula for the volume of a cone is V = (1/3)πr²h.
   Full step-by-step solution

   When a cone is cut by a plane parallel to its base, the resulting smaller cone is similar to the original cone. This similarity allows us to find the radius of the smaller cone using proportional relationships. The volume of a frustum is calculated as the difference between the volumes of two similar cones. The formula for the volume of a cone is V = (1/3)πr²h.

5. A pharmaceutical company is testing a new medication and wants to estimate what percentage of patients will experience a specific side effect. In a clinical trial with 500 randomly selected patients, 18% reported experiencing the side effect. Construct a 90% confidence interval for the true population proportion of patients who will experience this side effect. (Use z* = 1.645 for the critical value) Answer: 0.1518 to 0.2082 Solution: Identify the sample proportion p-hat = 0.18 and sample size n = 500 Calculate the standard error: SE = sqrt(p-hat(1-p-hat)/n) = sqrt(0.18*0.82/500) = sqrt(0.1476/500) = sqrt(0.0002952) = 0.01718 Calculate the margin of error: ME = z* × SE = 1.645 × 0.01718 = 0.02826 Construct the confidence…
   Full step-by-step solution

   Step 1: Identify the sample proportion p-hat = 0.18 and sample size n = 500 Step 2: Calculate the standard error: SE = sqrt(p-hat(1-p-hat)/n) = sqrt(0.18*0.82/500) = sqrt(0.1476/500) = sqrt(0.0002952) = 0.01718 Step 3: Calculate the margin of error: ME = z* × SE = 1.645 × 0.01718 = 0.02826 Step 4: Construct the confidence interval: p-hat ± ME = 0.18 ± 0.02826 Step 5: Lower bound = 0.18 - 0.02826 = 0.15174 Step 6: Upper bound = 0.18 + 0.02826 = 0.20826 Step 7: Round to four decimal places: 0.1518 to 0.2082 The confidence interval is 0.1518 to 0.2082.

6. Mere's exam scores are normally distributed with μ=78 and σ=12. Estimate the percentage of scores between 66 and 90. Answer: 68 Solution: Calculate the z-score for 66: z = (66 - 78)/12 = -12/12 = -1 Calculate the z-score for 90: z = (90 - 78)/12 = 12/12 = 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) Therefore,…
   Full step-by-step solution

   Step 1: Calculate the z-score for 66: z = (66 - 78)/12 = -12/12 = -1 Step 2: Calculate the z-score for 90: z = (90 - 78)/12 = 12/12 = 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 Mere's exam scores fall between 66 and 90. The answer is 68.

7. Liam's reaction times are normally distributed with μ=240 ms and σ=30 ms. Estimate the percentage of reaction times between 210 ms and 270 ms. Answer: 68 Solution: Calculate the z-score for 210 ms: z = (210 - 240)/30 = -30/30 = -1. Calculate the z-score for 270 ms: z = (270 - 240)/30 = 30/30 = 1.
   Full step-by-step solution

   Step 1: Calculate the z-score for 210 ms: z = (210 - 240)/30 = -30/30 = -1. Step 2: Calculate the z-score for 270 ms: z = (270 - 240)/30 = 30/30 = 1. Step 3: The empirical rule states that for a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean (between z = -1 and z = 1). Step 4: Therefore, approximately 68% of reaction times are between 210 ms and 270 ms. The answer is 68.