a) The average height of sunflowers in a field is 64 inwith 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 meanDescribe your drawing in as much detail as possible, and explain how you came up with each of your labels. b) there are 3,000 plants in the field, approximately how many will be taller than 71 inExplain how you got your answer .
a)
To draw a normal curve for the distribution of sunflower heights, start by drawing a horizontal axis labeled with the height values and standard deviations. The mean height of 64 in would be at the center of the axis. Calculate one, two, and three standard deviations from the mean by adding or subtracting 3.5 in each time.
- One standard deviation above the mean would be 64 + 3.5 = 67.5 in and below the mean would be 64 - 3.5 = 60.5 in.
- Two standard deviations above the mean would be 64 + 2(3.5) = 71 in and below the mean would be 64 - 2(3.5) = 57 in.
- Three standard deviations above the mean would be 64 + 3(3.5) = 74.5 in and below the mean would be 64 - 3(3.5) = 53 in.
Using these values, draw a smooth bell-shaped curve representing the distribution of sunflower heights.
b) To find out how many plants will be taller than 71 in, we can use the empirical rule for a normal distribution. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
Given that the mean height is 64 in and the standard deviation is 3.5 in, the height of 71 in is within two standard deviations above the mean.
Since 95% of the data falls within two standard deviations of the mean, we can expect approximately 95% of the 3,000 plants (or 0.95 * 3,000 = 2,850 plants) to have heights less than or equal to 71 in. Therefore, about 150 plants would be taller than 71 in in the field.