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The margin of error (E) is a statistic expressing the amount of random sampling error in a survey's results. Essentially, it tells us how much we can expect the sample estimate to vary from the actual population value. To calculate the margin of error, we use the formula: \[ E = Z_{\frac{α}{2}} \frac{s}{\sqrt{n}} \] In this formula, \( Z_{\frac{α}{2}} \) is the z-score corresponding to the desired confidence level (1.96 for 95% confidence), \( s \) is the sample standard deviation, and \( n \) is the sample size. For example, using our data: \[ s = 0.0518 \mathrm{g}, n = 100, Z_{0.025}=1.96 \] We get: \[ E = 1.96 \frac{0.0518}{\sqrt{100}} \approx 0.0101 \mathrm{g} \] By incorporating this margin of error into our confidence interval, we gain insight into how much our sample mean \( \bar{x} \) might reasonably diverge from the true population mean.

A confidence interval gives a range of values for a population parameter, based on a sample statistic. Specifically, it provides an estimated range of values which is likely to include the unknown population parameter at a given confidence level. To construct a confidence interval for the sample mean \( \bar{x} \), we use the margin of error (E): \[ \bar{x} - E < \mu < \bar{x} + E \] Substituting the values from our example: \[ \bar{x} = 0.8565 \mathrm{g}, E = 0.0101 \mathrm{g} \] We get: \[ 0.8565 - 0.0101 < \mu < 0.8565 + 0.0101 \] \[ 0.8464 < \mu < 0.8666 \] This is our 95% confidence interval for the population mean assuming a large population size. When dealing with finite populations where the sample size is more than 5% of the population size \( (n > 0.05N) \), we should use the finite population correction factor to adjust the margin of error.

Sample size refers to the number of observations in a sample. It is a critical component in the calculation of statistical estimates such as the margin of error and confidence intervals. Larger sample sizes typically result in more precise estimates of the population parameters. The finite population correction factor is used when dealing with a finite population. It corrects the standard error to account for the sampling without replacement by the following formula: \[ \sqrt{\frac{N-n}{N-1}} \] Where \( N \) is the population size and \( n \) is the sample size. For our exercise, \( N = 465 \) and \( n = 100 \). The finite population correction factor thus becomes: \[ \sqrt{\frac{465-100}{465-1}} \approx 0.937 \] By incorporating this factor, the corrected margin of error becomes: \[ E_{corrected} = E \times 0.937 \approx 0.0095 \mathrm{g} \] And the corrected confidence interval is: \[ 0.8470 < \mu < 0.8660 \] Overall, using the finite population correction factor makes our estimate more precise.