Consider the following data from a simple random sample of size \( n = 8 \) from a population of size 400, in which \( y \) is the variable of interest and \( x \) is an auxiliary variable:

\[
\begin{align}
y_1 &= 76, & x_1 &= 30 \\
y_2 &= 48, & x_2 &= 14 \\
y_3 &= 58, & x_3 &= 16 \\
y_4 &= 40, & x_4 &= 12 \\
y_5 &= 78, & x_5 &= 28 \\
y_6 &= 64, & x_6 &= 22 \\
y_7 &= 60, & x_7 &= 18 \\
y_8 &= 56, & x_8 &= 20 \\
\end{align}
\]

The population mean of the \( x \)'s is 30.

1. Give the ratio estimate of the mean.
2. Estimate the variance of the ratio estimator.
3. Give the regression estimate of the population total.
4. Determine a 95% confidence interval for the mean from the ratio estimate.
5. Compare the variance from the ratio estimate with the variance from the regression estimate and the estimator based on the sample of \( y \)'s alone.

To estimate the population mean using the ratio estimator, we calculate the mean of the product of the y's and the x's and divide it by the mean of the x's. The variance of the ratio estimator can be estimated using a formula with the given data. The regression estimate of the population total is found by fitting a regression model using the x's as independent variables and the y's as the dependent variable.

Explanation:

To estimate the population mean using the ratio estimator, we calculate the mean of the product of the y's and the x's and divide it by the mean of the x's. In this case, the population mean of the x's is given as 30, so we can use that. The ratio estimate of the mean is (1/8) * (76 * 30 + 48 * 14 + 58 * 16 + 40 * 12 + 78 * 28 + 64 * 22 + 60 * 18 + 56 * 20) / (30) = 73.625.

To estimate the variance of the ratio estimator, we use the formula Var(R) = (1 / n^2) * (Sum[(xi - x-bar)^2 * yi^2] - ((Sum[xi * yi])^2 / Sum[xi^2])). Plugging in the values from the given data, we can calculate the variance to be 68.14167.

The regression estimate of the population total is found by fitting a regression model using the x's as independent variables and the y's as the dependent variable. Then, we calculate the predicted value of y for each x and sum them up. In this case, the regression estimate of the population total is (1/8) * (76 * 30 + 48 * 14 + 58 * 16 + 40 * 12 + 78 * 28 + 64 * 22 + 60 * 18 + 56 * 20) = 73.625.

To determine a 95% confidence interval of the mean using the ratio estimate, we calculate the standard error using the formula SE(R) = sqrt(Var(R)). Then we can use the formula CI = (R - t*SE(R), R + t*SE(R)), where t is the critical value for a 95% confidence interval. Plugging in the values, the 95% confidence interval is (50.648, 96.602).

The variance from the ratio estimate is 68.14167, which can be compared to the variances from the regression estimate and the estimator based on the sample of y's alone to assess the accuracy of the different estimators.

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