a) The average height of sunflowers in a field is 64 inches with a standard deviation of 3.5 inches. Describe a normal curve for the distribution, including the values on the horizontal axis at one, two, and three standard deviations from the mean.
b) If there are 3,000 plants in the field, approximately how many will be taller than 71 inches?
a) Do you mean above or below the mean or both?
Z = (score-mean)/SD
Insert Z scores (±1, ±2 and/or ±3) into above equation and calculate.
b) Same equation.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply by 3,000.
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a) To describe a normal curve for the distribution, you need to consider the mean and standard deviation of the data. In this case, the average height of sunflowers is 64 inches with a standard deviation of 3.5 inches.
A normal curve, also known as a bell curve or Gaussian distribution, is a symmetrical and bell-shaped distribution of data points. The mean is positioned at the center of the curve, representing the average value. The standard deviation determines the spread of the data around the mean.
To find the values on the horizontal axis at one, two, and three standard deviations from the mean, you can follow these steps:
1. Find the values at one standard deviation from the mean:
- The lower value is obtained by subtracting one standard deviation from the mean: 64 - 3.5 = 60.5 inches.
- The upper value is obtained by adding one standard deviation to the mean: 64 + 3.5 = 67.5 inches.
2. Find the values at two standard deviations from the mean:
- The lower value is obtained by subtracting two standard deviations from the mean: 64 - (2 * 3.5) = 57 inches.
- The upper value is obtained by adding two standard deviations to the mean: 64 + (2 * 3.5) = 70 inches.
3. Find the values at three standard deviations from the mean:
- The lower value is obtained by subtracting three standard deviations from the mean: 64 - (3 * 3.5) = 53.5 inches.
- The upper value is obtained by adding three standard deviations to the mean: 64 + (3 * 3.5) = 70.5 inches.
So, the values on the horizontal axis at one, two, and three standard deviations from the mean are as follows:
- At one standard deviation: 60.5 inches to 67.5 inches.
- At two standard deviations: 57 inches to 70 inches.
- At three standard deviations: 53.5 inches to 70.5 inches.
b) To estimate the number of plants taller than 71 inches, we can use the concept of the normal distribution.
1. Calculate the z-score: First, we need to standardize the value of 71 inches using the z-score formula.
z = (x - mean) / standard deviation
z = (71 - 64) / 3.5
z ≈ 2
2. Find the proportion of plants taller than 71 inches: Using a standard normal distribution table or a calculator, we can find the proportion of data points beyond 2 standard deviations (which corresponds to the z-score of 2).
From the standard normal distribution table, the proportion beyond z = 2 is approximately 0.0228. This means, approximately 0.0228 or 2.28% of the plants will be taller than 71 inches.
3. Calculate the number of plants: Multiply the proportion from step 2 by the total number of plants in the field.
Number of plants taller than 71 inches ≈ 0.0228 * 3000 ≈ 68.4
Therefore, approximately 68 plants will be taller than 71 inches.