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 mean of 64 in. and should be symmetrical on both sides.
- The first standard deviation above and below the mean would be at 64 + 3.5 = 67.5 in. and 64 - 3.5 = 60.5 in. respectively.
- The second standard deviation above and below the mean would be at 64 + 2(3.5) = 71 in. and 64 - 2(3.5) = 57 in. respectively.
- The third standard deviation above and below the mean would be at 64 + 3(3.5) = 74.5 in. and 64 - 3(3.5) = 53.5 in. respectively.
b)
To find out how many plants will be taller than 71 in., we need to find the proportion of plants that are taller than 71 in. This can be done using the Z-score formula: Z = (X - mean) / standard deviation.
For X = 71 in., Z = (71 - 64) / 3.5 ≈ 2.
Looking at the standard normal distribution table, a Z-score of 2 corresponds to a probability of approximately 0.9772. This means that approximately 97.72% of the plants will be shorter than 71 in.
So, out of 3,000 plants, the approximate number of plants taller than 71 in. would be:
0.0278 * 3000 ≈ 83.4 plants
Therefore, approximately 83 plants will be taller than 71 in.