[1] 1.984217MATH 313: Survey Design and Sampling
Estimator of the Population Mean
Estimator of the Population Total
\[ \text{B} = t_{1-\frac{\alpha}{2}, n-1} \cdot \sqrt{\left(1-\frac{n}{N}\right) \cdot \frac{s^2}{n}} \]
\[ \text{B} = t_{1-\frac{\alpha}{2}, n-1} \cdot \sqrt{N^2 \cdot \left(1-\frac{n}{N}\right) \cdot \frac{s^2}{n}} \]
This function computes the critical value \(t_{1-\alpha / 2, n-1}\) for the t-distribution.
\[\begin{align*} \hat{\theta} \pm B \end{align*}\]
Here, \(\hat{\theta}\) is the estimate for the parameter of interest and \(B\) represents the bound on the error of estimation, computed as outlined in previous slides.
Example 1: A simple random sample of 100 houses within a community is selected and their monthly energy usages are recorded. The sample mean and sample standard deviation are founded to be \(\bar{y}=\) 887 kWh and \(s=120 \mathrm{kWh}\). If there are \(N=8000\) houses within the community. Estimate the true mean monthly energy usage per house, \(\mu\), and the total monthly energy usages for the community, \(\tau\). The estimation should include the estimator, the estimated variance, the \(95 \%\) bound on the error of estimation, and the \(95 \%\) confidence interval for the population parameter.
Mean Estimation Error
\[ B_{\bar{y}} = z_{1-\frac{\alpha}{2}} \cdot \sqrt{\left(\frac{N-n}{N-1}\right) \cdot \frac{\sigma^2}{n}} \]
Total Estimation Error \[ B_{\hat{\tau}} = z_{1-\frac{\alpha}{2}} \cdot N \cdot \sqrt{\left(\frac{N-n}{N-1}\right) \cdot \frac{\sigma^2}{n}} \]
Select \(B\) based on desired confidence and precision. Solve for \(n\) given \(B\), using the error bound formulas.
Case 1: Known Population Variance \(\sigma^2\)
\[ n = \frac{N \sigma^2}{(N-1) \left(\frac{B_{\bar{y}}}{z_{1-\frac{\alpha}{2}}}\right)^2 + \sigma^2} \]
Case 2: Known Sample Variance \(s^2\) from Previous Experiment
\[ n = \frac{N s^2}{N \left(\frac{B_{\bar{y}}}{z_{1-\frac{\alpha}{2}}}\right)^2 + s^2} \]
Case 1: Known Population Variance \(\sigma^2\) \[ n = \frac{N \sigma^2}{(N-1) \left(\frac{B_{\hat{\tau}}}{z_{1-\frac{\alpha}{2}} \cdot N}\right)^2 + \sigma^2} \]
Case 2: Known Sample Variance \(s^2\) from Previous Experiment \[ n = \frac{N s^2}{N \left(\frac{B_{\hat{\tau}}}{z_{1-\frac{\alpha}{2}} \cdot N}\right)^2 + s^2} \]
If \(\sigma^2\) and \(s^2\) are unknown, but the range \(R\) is given: Use equation for \(\sigma^2\) but approximate \(\sigma \approx \frac{R}{4}\).
Example 2: To study the monthly energy usages in a community with \(N=8000\) houses, a researcher want to select a simple random sample with size \(n\). Historical data shows that the sample standard deviation of the the monthly energy usages in this community is \(s=120 \mathrm{kWh}\). Find the sample size needed to estimate the average usage \(\mu\) with a bound on the error of estimation \(B_{\bar{y}}=50 \mathrm{kWh}\). How about the sample size required for estimate total usage \(\tau\) with \(B_{\hat{\tau}}=160000\)?
Example 3: The average amount of money \(\mu\) for a hospital’s accounts receivable must be estimated. Although no prior data are available to estimate the population variance, it is known that most accounts lie within a \(\$ 100\) range. There are \(N=1000\) open accounts. Find the sample size needed to estimate \(\mu\) with a bound on the error of estimation \(B_{\bar{y}}=\$ 3\).
\[\begin{align*} & \text{Starting with the margin of error equation:} \\ & B_{\bar{y}} = z_{1-\frac{\alpha}{2}} \cdot \sqrt{\left(\frac{N-n}{N-1}\right) \cdot \frac{\sigma^2}{n}} \\ & \text{Squaring both sides and isolating } n: \\ & \left(\frac{B_{\bar{y}}}{z_{1-\frac{\alpha}{2}}}\right)^2 = \frac{N-n}{N-1} \cdot \frac{\sigma^2}{n} \\ & \text{Cross-multiplying and rearranging for } n: \\ & (N-1) \left(\frac{B_{\bar{y}}^2}{z_{1-\frac{\alpha}{2}}^2 \sigma^2}\right) \cdot n + n = N \\ & n \left( (N-1) \left(\frac{B_{\bar{y}}^2}{z_{1-\frac{\alpha}{2}}^2 \sigma^2}\right) + 1 \right) = N \\ & n = \frac{N}{(N-1) \left(\frac{B_{\bar{y}}^2}{z_{1-\frac{\alpha}{2}}^2 \sigma^2}\right) + 1} \end{align*}\]
\[\begin{align*} & \text{Given the margin of error for unknown variance:} \\ & B_{\bar{y}} = t_{1-\frac{\alpha}{2}, n-1} \cdot \sqrt{\left(1-\frac{n}{N}\right) \cdot \frac{s^2}{n}} \\ & \text{Simplify and square both sides to isolate } n: \\ & \frac{B_{\bar{y}}}{t_{1-\frac{\alpha}{2}, n-1}} = \sqrt{\frac{N-n}{N} \cdot \frac{s^2}{n}} \\ & \left(\frac{B_{\bar{y}}}{t_{1-\frac{\alpha}{2}, n-1}}\right)^2 = \frac{N-n}{N} \cdot \frac{s^2}{n} \\ & \text{Rearrange to solve for } n: \\ & n \left(\frac{B_{\bar{y}}^2}{t_{1-\frac{\alpha}{2}, n-1}^2} + \frac{s^2}{N}\right) = s^2 \\ & n = \frac{s^2}{\frac{B_{\bar{y}}^2}{t_{1-\frac{\alpha}{2}, n-1}^2} + \frac{s^2}{N}} \end{align*}\]