a) The average height of sunflowers in a field is 64 in. with a standard deviation of 3.5 in. On a piece of paper, draw a normal curve for the distribution, including the values on the horizontal axis at one, two, and three
standard deviations from the mean. Describe your drawing in as much detail as possible, and explain how you came up with each of your labels.
b) If there are 3,000 plants in the field, approximately how many will be taller than 71 in.? Explain how you got your answer.
a) The normal curve should be centered at the average height of 64 in. The horizontal axis represents the heights of the sunflowers, with the mean of 64 in. at the center. One standard deviation above and below the mean would be 64 + 3.5 = 67.5 in. and 64 - 3.5 = 60.5 in. respectively. Two standard deviations above and below the mean would be 64 + 2(3.5) = 71 in. and 64 - 2(3.5) = 57 in. respectively. Three standard deviations above and below the mean would be 64 + 3(3.5) = 74.5 in. and 64 - 3(3.5) = 53.5 in. respectively. These values should be labeled on the horizontal axis of the normal curve.
b) To find out how many plants will be taller than 71 in., you need to find the proportion of plants that fall above 71 in. in the normal curve. First, you need to find the Z-score for a height of 71 in. Using the formula Z = (X - μ) / σ, where X is the height, μ is the mean, and σ is the standard deviation, you get Z = (71 - 64) / 3.5 = 2. To find the proportion of plants taller than 71 in., you can look up the corresponding proportion in a standard normal distribution table, which is approximately 0.0228.
To find the actual number of plants taller than 71 in., you would multiply the proportion by the total number of plants in the field, which is 0.0228 * 3000 = 68.4. Since you can't have a fraction of a plant, you would round up to estimate that approximately 69 plants will be taller than 71 in.