91Ó°ÊÓ

A confidence interval provides an estimated range of values which is likely to include an unknown population parameter. In this exercise, we focus on constructing a 99% confidence interval for the difference between two population means. This means we are 99% confident that the true difference between the population means falls within this interval.

To construct the confidence interval, we need the difference of the sample means, the standard error, and the t-critical value from the t-distribution. The confidence interval is calculated using the formula:

• Difference of means ± t-critical value × standard error

This formula indicates how much the sample means could vary to give us a true estimate of the population mean difference. If the interval does not include zero, it suggests a significant difference between the population means.

Understanding confidence intervals is crucial because it helps us make decisions based on data, showing whether differences observed in sample data might also be present in the general population.

The population mean is an average of a set of data points from the entire group being studied. In situations where collecting data from an entire population is impractical, we use samples to estimate the population mean.

In this exercise,

• The sample mean of the first population ( ar{x}_{1} ) is 125.2
• The sample mean of the second population ( ar{x}_{2} ) is 123.7

These sample means are used as our best estimates of the unknown population means. By comparing their difference, we aim to estimate the difference between the actual population means. Such comparisons are at the heart of inferential statistics, as they allow us to draw conclusions about populations based on sample data.

It’s essential to consider that sample means provide estimates which may have a margin of error. This is why the concept of confidence intervals and standard error become relevant, as they allow us to understand and account for this variability.

Pooled variance is a method used to estimate the variance of two or more populations to provide a more precise estimate when these populations' true variances are assumed equal. In this context, it's used because the two samples are independently and randomly selected, both having equal sizes (100).

To calculate the pooled variance (s_p^2), the formula is:

\[s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}\]
This formula combines the variances of both samples, using them to provide a single estimate that accounts for the slight differences in their individual variances. The pooled variance is then used to calculate the standard error, a key component in constructing confidence intervals.

Understanding pooled variance is essential when comparing means, as it ensures more accurate results by considering data from both samples together rather than individually.

The standard error of the sample means is a crucial statistical value used to measure how far the sample mean of the data is likely to be from the true population mean. In this exercise, it helps in determining how different the samples' mean differences may be.

• The formula for standard error when dealing with pooled variance is:

\[SE = \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}\]
This calculation gives an idea of the variability of sample means considered together, indicating how accurately the means of two samples reflect their populations.

A smaller standard error suggests that the estimate of the difference between population means is precise and reliable, enhancing the confidence in the conclusions drawn. Hence, when constructing a confidence interval, a smaller standard error directly leads to a narrower interval, suggesting less uncertainty about the estimate.