Select simple random samples of sizes 20 and 25 from the yield data of wheat crops from 100 villages, estimate the average yields and standard errors for both samples, and construc... Select simple random samples of sizes 20 and 25 from the yield data of wheat crops from 100 villages, estimate the average yields and standard errors for both samples, and construct 95% confidence intervals. Comment on the results.
Understand the Problem
The question relates to a statistical experiment involving the yield of wheat crops among a sample of villages. It asks to select random samples, estimate averages, and determine confidence intervals, likely for practical applications in agricultural research.
Answer
The confidence interval for the average yield at a 95% level is given by $(\bar{x} - ME, \bar{x} + ME)$, calculated based on selected samples.
Answer for screen readers
The confidence interval at 95% for the average yield based on the selected sample can be denoted as:
$$ (\bar{x} - ME, \bar{x} + ME) $$
The specific numerical values would depend on the actual calculations.
Steps to Solve
- Select Random Samples
Choose a simple random sample from the provided data. For this example, let's say we select 20 values for the first part.
- Calculate the Average Yield
To find the average yield, add all selected samples and divide by the number of samples (20).
$$ \text{Average} = \frac{\text{Sum of samples}}{\text{Number of samples}} $$
- Calculate the Standard Deviation
Use the following formula to calculate the standard deviation (SD) of the selected samples.
$$ SD = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} $$
where ( \bar{x} ) is the mean, and ( n ) is the number of samples.
- Determine the Standard Error
Calculate the standard error (SE) using the formula:
$$ SE = \frac{SD}{\sqrt{n}} $$
where ( n ) is the size of the sample.
- Find the z-value for a 95% Confidence Interval
For a 95% confidence interval, the z-value is approximately 1.96.
- Calculate the Margin of Error
Determine the margin of error (ME) using:
$$ ME = z \cdot SE $$
- Calculate the Confidence Interval
Finally, the confidence interval is given by:
$$ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) $$
The confidence interval at 95% for the average yield based on the selected sample can be denoted as:
$$ (\bar{x} - ME, \bar{x} + ME) $$
The specific numerical values would depend on the actual calculations.
More Information
The confidence interval provides a range within which we can expect the true average yield to fall, based on our sample data. This is crucial in agricultural research to make informed decisions for crop management.
Tips
- Not Randomizing the Sample: Ensure that the sample is randomly selected to avoid bias.
- Incorrect Calculation of Standard Deviation: Pay attention to the formula and double-check calculations.
- Confusing Standard Deviation with Standard Error: Remember that standard error is derived from standard deviation but accounts for sample size.
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