Module Six Lesson Four Math

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Smithfield-selma High *

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Statistics

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Nov 24, 2024

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Given that the heights of males ages 18-25 (measured in inches) have the distribution N(69.5, 1.5), find each of the following. This notation means that it follows a normal curve, their mean height is 69.5 inches, and their standard deviation is 1.5 inches. Draw a normal curve first and use that to answer the questions. Make sure that you label the x-axis correctly. You will need to find z-scores to answer each of these questions. 1. How tall would an 18 to 25 male have to be in order to be in the 73rd percentile? We need to find the closest value to 0.73 in the table, which is 0.7324. The z-score corresponding to this value is 0.62. In this case, the mean height is 69.5 inches and the standard deviation is 1.5 inches. Plugging in the values, we have: 0.62 = (x - 69.5) / 1.5 Solving for x, we get: 0.62 * 1.5 = x - 69.5 0.93 = x - 69.5 x = 0.93 + 69.5 x = 70.43 Therefore, an 18 to 25-year-old male would need to be approximately 70.43 inches tall in order to be in the 73rd percentile.
2. What proportion of males 18 to 25 are between 6 8 and 71 inches tall? To determine the proportion of males 18 to 25 who are between 68 and 71 inches tall, we can use the z-score formula again. We calculate the z-scores for the lower and upper bounds of the range: Lower bound: z = (68 - 69.5) / 1.5 z = -1 Upper bound: z = (71 - 69.5) / 1.5 z = 1 The z-score of -1 corresponds to a cumulative probability of 0.1587, and the z-score of 1 corresponds to a cumulative probability of 0.8413. To find the proportion between these two z-scores, we subtract the lower cumulative probability from the upper cumulative probability: 0.8413 - 0.1587 = 0.6826 Therefore, approximately 68.26% of males 18 to 25 are between 68 to 71 inches tall 3. If a 20-year-old male is 73 inches tall, what percentile would this place him? (73 - 69.5) / 1.5 = 2.33 z-score of 2.33 = 0.9901 He would be in the 99th percentile. 4. What is the probability that a randomly selected male (age 18 to 25) is less than 67 inches tall or over 72 inches tall? (72 - 69.5) / 1.5 = 1.67 z-score of 1.67 = 0.9525 (67 - 69.5) / 1.5 = -1.67 z-score of -1.67 = 0.0475 0.9525 - 0.0475 = 0.905 1.00 - 0.905 = 0.095 = 9.5% chance he is less than 67 inches tall or over 72 inches tall. 5. What is the percentile for an 18-25 year old male that is 66 inches tall? Is the height of 66 inches unusual for an 18-25 year old male? Justify your answer. (66 - 69.5) / 1.5 = -2.33 Z score is 0.099 = 1% He is in the 1st percentile which is unusual as 99% of men are taller than 66 inches
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