Question

Calculate the margin of error in estimating a population mean \(\mu\) for the values given in Exercises 3-6. Comment on how a larger population variance affects the margin of error: \(n=30, \sigma^{2}=1.5\)

Short answer

Answer: A larger population variance leads to a larger margin of error, making the estimate of the population mean less precise.

Step-by-step solution

Calculate the population standard deviation

The population standard deviation is the square root of the population variance: \(\sigma = \sqrt{\sigma^2}\). In our case, \(\sigma = \sqrt{1.5}\).

Determine the critical value for the desired level of confidence

Since we are not given a specific level of confidence, we will assume a 95% confidence level. The critical value (z-score) for a 95% confidence level is 1.96.

Calculate the margin of error

The formula for the margin of error is \(ME = z \cdot \frac{\sigma}{\sqrt{n}}\). Plugging in the values, we have: \(ME = 1.96 \cdot \frac{\sqrt{1.5}}{\sqrt{30}} \approx 0.5545\) So, the margin of error is approximately 0.5545.

Comment on how a larger population variance affects the margin of error

Population variance is directly proportional to the margin of error. The higher the variance, the more it affects the margin of error by making it larger. This implies that the estimate will be less precise when the population variance is higher since there is a greater spread in the values. Therefore, if we were to have a larger population variance than 1.5, the margin of error would be even larger than 0.5545, which would lead to a less precise estimate of the population mean.

Key concepts

Population Mean Estimation

When statisticians set out to estimate the population mean \(\mu\) - the average value of a trait across an entire population - they often work with samples, due to the impracticalities of surveying everyone in a population. To estimate this mean as accurately as possible, a sample mean \(\bar{x}\) is calculated, which serves as the best guess for the population mean. However, it's important to recognize that due to random sampling errors, the sample mean may not exactly match the actual population mean.

To account for these potential discrepancies, statisticians calculate the margin of error (ME), which reflects how much the estimate \(\bar{x}\) might deviate from the true \(\mu\). Understanding the margin of error aids in interpreting the precision of an estimate. A small margin of error suggests that the sample mean is likely very close to the actual population mean, while a large margin means the estimate might be less reliable.

Population Variance

Population variance \(\sigma^2\) measures how much the values within a population diverge from the mean. Variance is important because it provides a numerical value for the spread of a dataset – the higher the variance, the more spread out the individual data points are. It is calculated by taking the average of the squared differences between each data value and the population mean.

The square root of the population variance, known as the population standard deviation \(\sigma\), is often used in practice because it keeps the same unit as the data, making interpretation more straightforward. Variance is crucial in calculating the margin of error since higher variance implies that individual data points tend to be farther from the mean, which increases the potential for the sample mean to be different from the population mean.

Confidence Interval

A confidence interval (CI) provides a range of values that likely contain the true population mean with a certain level of confidence. Typically expressed in percentage, such as 95%, the confidence level represents how certain we are that the interval includes the true mean, should we replicate the sampling process multiple times.

The formula to calculate the confidence interval around a sample mean includes the margin of error: \[CI = \bar{x} \pm ME\]. The plus-minus symbol represents the range above and below the sample mean where the population mean is expected to fall. Confidence intervals are especially helpful for conveying the reliability of statistical estimates, as they offer a way to visually assess the precision and stability of the sample mean in capturing the population mean.