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.
(3 points)
a) To draw the normal curve for the distribution of sunflower heights, first, we plot the mean height of 64 inches in the middle of the horizontal axis. Next, we mark one standard deviation above and below the mean, which would be 64 + 3.5 = 67.5 inches and 64 - 3.5 = 60.5 inches. These values represent approximately 68% of the data.
Then, we mark two standard deviations above and below the mean, which would be at 64 + 2(3.5) = 71 inches and 64 - 2(3.5) = 57 inches. These values represent approximately 95% of the data.
Finally, we mark three standard deviations above and below the mean, which would be at 64 + 3(3.5) = 74.5 inches and 64 - 3(3.5) = 55.5 inches. These values represent approximately 99.7% of the data.
b) To estimate the number of plants taller than 71 inches in a field of 3,000 plants, we can use the empirical rule for a normal distribution. According to the empirical rule, approximately 2.5% of the data falls three standard deviations above the mean.
So, out of 3,000 plants, approximately 0.025 * 3,000 = 75 plants will be taller than 71 inches. Therefore, approximately 75 plants in the field will be taller than 71 inches.