91Ó°ÊÓ

The Z-score is a critical statistic often used in constructing confidence intervals. It measures the number of standard deviations a data point lies from the mean. For a confidence interval, the Z-score helps determine the range where the true population mean is expected to lie.

• For a 95% confidence interval, the Z-score is approximately 1.96. This means that 95% of the sample means will fall within 1.96 standard deviations of the true mean under the normal distribution.
• When reducing the confidence level to 90%, the Z-score decreases to about 1.645. This indicates a smaller range, as only 90% of the data is captured.

The Z-score is crucial as it directly affects the width of the confidence interval. A higher Z-score indicates a wider interval, providing greater certainty that the true mean lies within this range, whereas a lower Z-score results in a narrower interval.

The error bound is a vital component in the calculation of a confidence interval. It represents the margin of error around the sample mean. The formula for the error bound (\(E\)) is:\[E = Z \times \frac{\sigma}{\sqrt{n}}\],
where:

• \(Z\) is the Z-score,
• \(\sigma\) is the standard deviation,
• \(n\) is the sample size.

Changing the confidence level affects the Z-score, hence influencing the error bound. For example, at a 95% confidence level, the error bound is larger due to the Z-score of 1.96. However, decreasing the confidence level to 90% reduces the Z-score to 1.645, resulting in a smaller error bound.

This change illustrates a trade-off: higher confidence levels provide broader intervals with larger error bounds, offering more assurance that the true mean falls within the interval. Conversely, lower confidence levels produce a narrower interval with a smaller error bound, offering less certainty.

The sample mean is a fundamental statistic used to estimate the population mean. It is the average value obtained from a sample, calculated by summing up all the sample data points and dividing by the number of data points.
For constructing a confidence interval, the sample mean serves as the midpoint.

• The confidence interval is built around this value, extending equally in both directions based on the error bound.
• As the sample size increases, the sample mean becomes a more reliable estimate of the population mean, due to the Central Limit Theorem suggesting that the distribution of the sample mean approaches a normal distribution.

The sample mean benefits from a larger sample size, which helps reduce the variability and make the estimate more precise. This is crucial for narrowing down the confidence interval, providing a more accurate reflection of the true population parameter.