The range for a set of data is estimated to be 24. a. What is the planning value for the population standard deviation? b. At 95% confidence, how large a sample would provide a margin of error of 3? c. At 95% confidence, how large a sample would provide a margin of error of 2?
The correct answer and explanation is:
To answer these questions, we’ll use statistical formulas for estimating the population standard deviation from the range, and then apply those to the margin of error formula for estimating sample size.
a. What is the planning value for the population standard deviation?
We estimate the standard deviation (σ) from the range (R) using the range rule of thumb: σ≈Range4\sigma \approx \frac{\text{Range}}{4}
Given that the range is 24: σ≈244=6\sigma \approx \frac{24}{4} = 6
✅ Answer: The planning value for the population standard deviation is 6.
b. At 95% confidence, how large a sample would provide a margin of error of 3?
To find the required sample size for a given margin of error (E) at 95% confidence, use the formula: n=(z⋅σE)2n = \left(\frac{z \cdot \sigma}{E}\right)^2
Where:
- zz = 1.96 for 95% confidence
- σ=6\sigma = 6
- E=3E = 3
n=(1.96⋅63)2=(11.763)2=(3.92)2≈15.37n = \left(\frac{1.96 \cdot 6}{3}\right)^2 = \left(\frac{11.76}{3}\right)^2 = (3.92)^2 \approx 15.37
Always round up for sample size:
✅ Answer: A sample size of 16 is needed.
c. At 95% confidence, how large a sample would provide a margin of error of 2?
Same formula, but with E=2E = 2: n=(1.96⋅62)2=(11.762)2=(5.88)2≈34.57n = \left(\frac{1.96 \cdot 6}{2}\right)^2 = \left(\frac{11.76}{2}\right)^2 = (5.88)^2 \approx 34.57
Round up:
✅ Answer: A sample size of 35 is needed.
Explanation (300 Words)
In statistics, the range rule of thumb provides a simple estimate of the population standard deviation. By dividing the range by 4, you approximate the typical spread of data, assuming a relatively normal distribution. This is useful when only limited data is available, such as an estimated range.
Once the standard deviation is estimated, we can determine the necessary sample size to achieve a desired margin of error (E) for a population mean. The margin of error represents how close a sample mean is expected to be to the true population mean, with a certain level of confidence.
Using the formula: n=(z⋅σE)2n = \left(\frac{z \cdot \sigma}{E}\right)^2
we plug in the z-score for 95% confidence, which is 1.96, a standard value from the normal distribution.
For a margin of error of 3, we need fewer observations because we’re willing to tolerate more error. But for a smaller error of 2, a larger sample is required to tighten the estimate around the true mean.
This process is a fundamental part of sample planning, ensuring that studies or surveys are neither underpowered (too few participants) nor unnecessarily large (wasting resources).