The seasonal output of a new experimental strain of pepper plants was carefully weighed. The mean weight per plant is 15.0 pounds, and the standard deviation of the normally distributed weights is 1.75 pounds. Of the 200 plants in the experiment, how many produced peppers weighing between 13 and 16 pounds?
a. 100
b. 118
c. 197
d. 53
I took the easy way out and used the Java normal distribution tool tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html
Enter the mean and the sigma and the higher and lower weight units and then click "enter". The probable fraction with weights in that interval is 0.5896
The answer (after multplying by 200 students) is 118
The same result could be obtained by using a table of the error function.
To find the number of pepper plants that produced peppers weighing between 13 and 16 pounds, we need to calculate the z-score for each weight and then find the proportion of plants within that range.
To calculate the z-score, we use the formula:
z = (x - μ) / σ
where x is the value of the weight, μ is the mean weight, and σ is the standard deviation.
For 13 pounds:
z = (13 - 15) / 1.75 = -1.14
For 16 pounds:
z = (16 - 15) / 1.75 = 0.57
To find the proportion of plants within this range, we need to find the area under the normal curve between these two z-scores. We can use a standard normal distribution table or a calculator to find this area.
Using a standard normal distribution table, we find that the area to the left of z = -1.14 is 0.1271 and the area to the left of z = 0.57 is 0.7157.
To find the proportion of plants within this range, we subtract the smaller area from the larger area:
0.7157 - 0.1271 = 0.5886
Therefore, approximately 58.86% or 0.5886 of the plants produced peppers weighing between 13 and 16 pounds.
Now, to find the actual number of plants, we multiply the proportion by the total number of plants:
0.5886 * 200 = 117.72
Since we can't have a fraction of a plant, we round the answer to the nearest whole number:
117.72 rounded to the nearest whole number is 118.
Therefore, the answer is b. 118 plants produced peppers weighing between 13 and 16 pounds.
To solve this problem, we need to calculate the number of plants that produced peppers weighing between 13 and 16 pounds.
First, we can convert these weights into z-scores by using the formula:
z = (x - μ) / σ
where:
x is the weight of the peppers,
μ is the mean weight per plant,
σ is the standard deviation of the weights.
For the lower weight of 13 pounds:
z1 = (13 - 15) / 1.75
z1 = -2 / 1.75
z1 ≈ -1.14
For the upper weight of 16 pounds:
z2 = (16 - 15) / 1.75
z2 = 1 / 1.75
z2 ≈ 0.57
Next, we can use a standard normal distribution table or a calculator to find the area under the curve between these z-scores.
For -1.14, the area is approximately 0.1271.
For 0.57, the area is approximately 0.7123.
The difference between these two areas represents the proportion of plants that produced peppers weighing between 13 and 16 pounds.
Proportion = 0.7123 - 0.1271
Proportion ≈ 0.5852
To find the number of plants, we multiply this proportion by the total number of plants in the experiment:
Number of plants = 0.5852 * 200
Number of plants ≈ 117.04
Therefore, the correct option is b. 118 plants produced peppers weighing between 13 and 16 pounds.