If 12 Men Can Do A Certain Work In 18 Days How Much Time Wou
If 12 Men Can Do A Certain Work In 18 Days How Much Time Would It Tak
If 12 men can complete a certain task in 18 days, the question asks how much time it would take for only 2 men to complete the same task. To solve this, we start by understanding that work done is inversely proportional to the number of workers when the work rate is constant.
Step 1: Determine total work in terms of man-days
Total work (W) can be expressed as the product of the number of men and the number of days they work:
\[ W = \text{Number of men} \times \text{Number of days} \]
Given data:
\[ W = 12 \text{ men} \times 18 \text{ days} = 216 \text{ man-days} \]
This total work remains constant regardless of how many men are working, assuming they work at the same rate.
Step 2: Calculate time for 2 men
Let \( x \) be the number of days 2 men need to complete the same work:
\[ 2 \text{ men} \times x \text{ days} = 216 \text{ man-days} \]
Solve for \( x \):
\[ x = \frac{216}{2} = 108 \text{ days} \]
Conclusion: It would take 2 men 108 days to complete the same work.
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Estimating the Minimum Sample Size for a Population Mean with a Margin of Error
The second part of the problem involves estimating the minimum sample size required to estimate the population mean house price within a specified margin of error, based on prior information from pilot studies.
Given Data:
- Margin of error (E): $10,000
- Population standard deviation (\( \sigma \)): $70,000
Objective:
Find the minimum sample size \( n \) such that the margin of error \( E \) is not exceeded when estimating the population mean.
Step 1: Recall the formula for the margin of error in a normal distribution
\[ E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
Where:
- \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Commonly, a 95% confidence level is used, for which:
\[ z_{0.025} \approx 1.96 \]
Step 2: Rearrange formula to solve for \( n \):
\[ n = \left( \frac{z_{\alpha/2} \times \sigma}{E} \right)^2 \]
Substituting the known values:
\[ n = \left( \frac{1.96 \times 70,000}{10,000} \right)^2 \]
Calculate numerator:
\[ 1.96 \times 70,000 = 137,200 \]
Divide numerator by margin of error:
\[ \frac{137,200}{10,000} = 13.72 \]
Square this value:
\[ n = (13.72)^2 \approx 188.26 \]
Since sample size must be an integer, we round up:
\[ n = 189 \]
Conclusion: A minimum sample size of 189 houses is needed to estimate the mean house price within a margin of error of $10,000 with 95% confidence.
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Summary
In solving the work problem, understanding the inverse relationship between workers and days allows us to determine the time needed for fewer workers. For the statistical problem, applying the margin of error formula for a mean with known standard deviation enables the estimation of the necessary sample size to achieve desired accuracy at a specified confidence level.
References
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