The average height of sunflowers in a field is 64 inches. With a standard deviation of 3.5 inches. 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 deviation from the mean. Describe your drawing in as much detail as possible and explain how you came up with each of your labels.
If there are 3000 pants in the field, approximately how many will be taller than 71 inches? Explain how you got your answer.
To draw the normal curve, we start by placing the mean at the center of the axis. In this case, the mean is 64 inches. Then, we draw a vertical line at the mean, which represents the highest point on the curve. We label this line "mean" or "μ".
Next, we add the standard deviations to the axis, which are represented as the number of units away from the mean. Since the standard deviation is 3.5 inches, we draw two lines, one at 60.5 inches (which is 1 standard deviation below the mean) and one at 67.5 inches (which is 1 standard deviation above the mean). We label these lines "+1σ" and "-1σ", respectively.
Next, we draw two more lines at 57 inches and 71 inches, which are 2 standard deviations below and above the mean, respectively. We label these lines "-2σ" and "+2σ". Finally, we draw two more lines at 53.5 inches and 74.5 inches, which are 3 standard deviations below and above the mean, respectively. We label these lines "-3σ" and "+3σ".
The resulting curve should be a bell-shaped curve that is symmetrical around the mean.
To estimate how many pants will be taller than 71 inches, we need to calculate the z-score for this value, which tells us how many standard deviations away from the mean 71 inches is. To do this, we use the formula:
z = (x - μ) / σ
where x is the value we're interested in (71 inches), μ is the mean (64 inches), and σ is the standard deviation (3.5 inches).
Plugging in the numbers, we get:
z = (71 - 64) / 3.5 = 2
This means that 71 inches is 2 standard deviations above the mean. Looking at our normal curve, we can see that the area to the right of +2σ represents the percentage of pants that are taller than 71 inches. Using a standard normal distribution table, we can find that this area is approximately 2.28%.
Therefore, out of 3000 pants, we can estimate that approximately 68.4 will be taller than 71 inches (2.28% of 3000).
To draw a normal curve for the distribution of sunflower heights, we will use the mean height of 64 inches and the standard deviation of 3.5 inches.
1. Start by drawing a horizontal axis labeled "Height (inches)" with a suitable scale. From left to right, label this axis with values starting from the mean and going both directions. The mean height of 64 inches should be at the center of the axis.
2. To the left of the mean, mark the first standard deviation (3.5 inches) lower than the mean, which is 60.5 inches. This represents one standard deviation below the mean.
3. To the right of the mean, mark the first standard deviation (3.5 inches) higher than the mean, which is 67.5 inches. This represents one standard deviation above the mean.
4. To find the values for two standard deviations below and above the mean, subtract or add two times the standard deviation (7 inches) to the mean. This gives us 57 inches (two standard deviations below the mean) and 71 inches (two standard deviations above the mean), respectively.
5. Finally, repeat the previous step to find the values for three standard deviations below and above the mean. Subtract or add three times the standard deviation (10.5 inches) to the mean, resulting in 53.5 inches (three standard deviations below the mean) and 74.5 inches (three standard deviations above the mean).
The area under the normal curve represents the probability of finding a sunflower at a specific height. In this case, we want to know the approximate number of sunflowers taller than 71 inches out of 3000 flowers.
To estimate this, we need to calculate the z-score (standard deviation units away from the mean) for 71 inches using the formula:
Z = (X - μ) / σ
Where X is the specific height (71 inches), μ is the mean (64 inches), and σ is the standard deviation (3.5 inches).
So, the z-score for 71 inches would be:
Z = (71 - 64) / 3.5 = 2
Using a standard normal distribution table or calculator, we can find that the area to the right of a z-score of 2 is approximately 0.0228.
Therefore, approximately 0.0228 (or 2.28%) of the sunflowers will be taller than 71 inches out of the total 3000 pants in the field.
To find the approximate number, we multiply the proportion by the total number of sunflowers:
0.0228 * 3000 = 68.4
Therefore, approximately 68 sunflowers will be taller than 71 inches out of the 3000 pants in the field.