Question 22·Easy·Inference from Sample Statistics and Margin of Error
A farmer owns an orchard with 1,200 identical apple trees. To estimate the total number of apples in the orchard, the farmer randomly selects eight trees and counts the apples on each. The counts are
120, 135, 110, 125, 130, 140, 115, and 125.
Based on the mean of this sample, which of the following ranges best estimates the total number of apples on the 1,200 trees?
For questions that use a sample to estimate a total, first compute the sample mean (add all sample values and divide by the number of samples), then multiply that mean by the total population size to estimate the overall total. Do the arithmetic in simple steps (e.g., group sums, use mental math shortcuts like multiplying by 12 and then by 100) and finally pick the answer choice whose range includes your estimate, checking that your choice is not obviously too low or too high compared with your calculation.
Hints
Think about what you can get from the sample
The farmer counted apples on 8 trees. Before you can estimate the total for 1,200 trees, what information about a typical tree should you find from the sample?
Compute the sample mean
Add up the 8 apple counts and divide by 8 to find the average number of apples per tree in the sample.
Scale the mean up to the whole orchard
Once you have the average apples per tree from the sample, multiply that average by 1,200 (the total number of trees). Then see which answer choice has a range that contains that estimate.
Desmos Guide
Use Desmos to find the sample mean
In Desmos, type the expression (120 + 135 + 110 + 125 + 130 + 140 + 115 + 125)/8 and press Enter. The output is the average number of apples per tree in the sample.
Use Desmos to estimate the orchard total
In a new line, type the expression (<previous_result>) * 1200 (or directly ((120 + 135 + 110 + 125 + 130 + 140 + 115 + 125)/8) * 1200). The output is the estimated total number of apples for all 1,200 trees. Compare this number to the ranges in the answer choices and choose the one that contains it.
Step-by-step Explanation
Understand what the sample tells you
The farmer used a random sample of 8 trees to estimate the average number of apples per tree in the whole orchard.
To estimate the total number of apples on all 1,200 trees, you should:
- First find the mean (average) apples per tree from the 8 sample trees.
- Then multiply that mean by 1,200 (the total number of trees).
Calculate the total apples in the sample and the sample mean
Add the apple counts from the 8 trees:
Group or add step by step:
So the total number of apples in the sample is 1,000.
Now find the mean (average) apples per tree by dividing by 8 (the number of trees in the sample):
So the sample mean is 125 apples per tree.
Estimate the total apples in the orchard
Use the sample mean to estimate the total apples on all 1,200 trees.
Multiply the mean apples per tree by 1,200 trees:
First multiply :
Now remember that , so:
So the estimate for the total number of apples in the orchard is 150,000.
Match the estimate to the best range
Your estimated total number of apples is 150,000. Now pick the answer choice whose range includes this value:
- A) 135,000 to 140,000 (too low)
- B) 140,000 to 145,000 (still too low)
- C) 145,000 to 155,000 (this includes 150,000)
- D) 160,000 to 170,000 (too high)
Therefore, the best estimate is 145,000 to 155,000, which is choice C.