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: \(n=30, \sigma^{2}=.9\)

Short answer

Answer: A larger population variance leads to a greater population standard deviation, which in turn will increase the margin of error. This is because a larger variance indicates a greater spread of the data around the population mean, making it more difficult to estimate the population mean with a high degree of precision. Therefore, the margin of error needs to be larger to account for this greater uncertainty.

Step-by-step solution

Calculate the population standard deviation

To find the population standard deviation \((\sigma)\), take the square root of the given variance \((\sigma^{2})\). In this case, the variance is 0.9. \(\sigma = \sqrt{0.9} \approx 0.95\)

Calculate the margin of error

Now, plug in the given values (\(n\), \(\sigma\), and \(z\)) into the margin of error formula: Margin of Error = \(z\frac{\sigma}{\sqrt{n}}\) For a 95% confidence level, \(z \approx 1.96\). Margin of Error = \(1.96\frac{0.95}{\sqrt{30}} \approx 0.66\)

Comment on the effect of a larger population variance

A larger population variance leads to a greater population standard deviation, which in turn will increase the margin of error. This is because a larger variance indicates a greater spread of the data around the population mean. As a result, it becomes more difficult to estimate the population mean with a high degree of precision, and the margin of error needs to be larger to account for this greater uncertainty.

Key concepts

Population Mean Estimation

Estimating the population mean is a fundamental concept in statistics, representing the average value of a variable within a population. It is denoted as \(\mu\). Since collecting data from an entire population is often impractical, we rely on sample means to estimate the population mean. To enhance the accuracy of our estimation, statisticians use the margin of error (MOE) which accounts for the potential discrepancy between the sample mean and the population mean.

The formula for calculating the margin of error incorporates the standard deviation of the population (\(\sigma\)) and the size of the sample (\(n\)), along with the z-score, a statistical metric that corresponds to the desired confidence level. Essentially, the margin of error increases as the sample standard deviation rises and decreases as the sample size grows. This is crucial because it helps in ensuring that the population mean is within a certain range, thus providing confidence in the results of the study.

Population Variance

Population variance (denoted as \(\sigma^{2}\)) measures the degree to which data points in a population deviate from the population mean. A high variance indicates that the data points are spread out widely around the mean, while a low variance suggests that the data points are closely clustered around the mean.

To understand the effect of population variance on our estimations, remember the fundamental property that the standard deviation is the square root of the variance (\(\sigma = \sqrt{\sigma^{2}}\)). The variance is a critical factor in margin of error calculations because it directly influences the population standard deviation. A larger population variance implies a larger standard deviation, which, as per the margin of error formula, would result in an increase in the margin of error. Therefore, understanding how variance works and its impact on statistical estimations is crucial for interpreting the results accurately.

Confidence Level

The confidence level is a statistical measure that expresses the degree of certainty in a sampling method. Expressed as a percentage (usually 90%, 95%, or 99%), the confidence level signifies how likely it is that the population mean will fall within the calculated margin of error of a sample mean.

A 95% confidence level, which is commonly used, means that if we were to take 100 different samples and compute their respective confidence intervals, we would expect about 95 of those intervals to contain the true population mean. To account for the desired confidence level in our calculations, we use a z-score, which is a measure of how many standard deviations an element is from the mean of a distribution. For a 95% confidence level, the z-score is approximately 1.96. This level of confidence is considered a standard balance between certainty and precision in many fields of research.