Estimate The Average Annual Expenses Of Students On Books
Estimate The Average Annual Expenses Of Students On Books
To estimate the average annual expenses of students on books and class materials, a sample of 36 students was taken. The sample mean was $850, with a standard deviation of $54. To construct a 99% confidence interval for the population mean, we employ the t-distribution because the population standard deviation is unknown and the sample size is less than 30. However, since the sample size is 36, which exceeds 30, the Z-distribution can be appropriately used for an approximate calculation.
The formula for the confidence interval is:
CI = sample mean ± Zα/2 * (sample standard deviation / √n)
Where Zα/2 is the Z-value corresponding to the desired confidence level, which for 99% is approximately 2.576.
Calculating the margin of error (ME):
ME = 2.576 (54 / √36) = 2.576 (54 / 6) = 2.576 * 9 = approximately $23.18.
Thus, the 99% confidence interval for the population mean expenses is:
$850 - $23.18 to $850 + $23.18, or approximately $826.82 to $873.18.
This interval indicates that we are 99% confident that the true average annual expenses of students on books fall within this range.
Paper For Above instruction
Estimating the research parameters such as mean expenses or transaction times necessitates the use of confidence intervals, which provide a range of plausible values for the population parameter based on sample data. In the context of the student expenses on books, using a sample of 36 students with a mean of $850 and a standard deviation of $54, the confidence interval calculation involves selecting an appropriate critical value and measuring the precision of the estimate.
The choice of a 99% confidence level maximizes the interval's reliability but requires a larger critical value. Because the sample size is relatively large (n=36), the Z-distribution can be applied for simplicity, with a Z-value of approximately 2.576 for 99% confidence. The margin of error calculated (\$23.18) reflects the degree of precision in estimating the true average.
Applying the formula, the computed confidence interval extends from approximately \$826.82 to \$873.18. This range indicates that, with high confidence, the true mean expenses of students on books are contained within these bounds. Such estimates are crucial for policymakers and educational institutions in planning budgets and financial aid structures.
Beyond simple point estimates, confidence intervals offer a robust statistical framework for inference, accommodating sampling variability and providing a range for potential economic planning. Properly interpreting these intervals ensures that decision-making is based on sound statistical underpinning, reducing risks associated with underestimating or overestimating true expenses.
In conclusion, statistical inference tools like confidence intervals are indispensable in educational financial assessments, and the methods applied in this example demonstrate how sample data can effectively inform understanding of broader population parameters.
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