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) To draw the normal curve for the distribution of sunflower heights, start by drawing a horizontal axis labeled "Height of Sunflowers (in.)" with the mean of 64 in. at the center. Then, mark one standard deviation above and below the mean at 67.5 in. and 60.5 in., two standard deviations above and below the mean at 71 in. and 57 in., and three standard deviations above and below the mean at 74.5 in. and 53.5 in.
Next, draw a bell-shaped curve that peaks at the mean of 64 in. and gradually tapers off towards the edges. Make sure the curve touches the horizontal axis at each of the labeled points to represent the percentage of sunflowers within each standard deviation range.
b) To find the number of plants taller than 71 in., we need to calculate the percentage of sunflowers that fall above this height based on the standard normal distribution.
First, we need to determine how many standard deviations above the mean 71 in. is. Since the mean is 64 in. and the standard deviation is 3.5 in., we can calculate:
(71 in. - 64 in.) / 3.5 in. = 2 standard deviations above the mean
From the standard normal distribution table, we know that about 2.28% of the distribution is above 2 standard deviations.
So, if there are 3,000 plants in the field, we can estimate that approximately 2.28% of 3,000 or about 68 sunflowers will be taller than 71 in.