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When comparing two separate groups to evaluate the outcome of an experiment or a survey, we often look at the **difference of means**. This metric represents how much the average of one group deviates from the average of another. For example, if you’re comparing the average heights of students from two different schools, the difference of means ( \( \mu_1 - \mu_2 \)) will show whether one school on average has taller students than the other.

A confidence interval for this difference helps us understand whether the observed difference is statistically significant or if it might be due to random chance. If the interval for the difference of means is entirely negative, as in our exercise, it indicates that statistically, the first group has consistently lower values than the second group.

In statistics, a **population parameter** is a value that gives information about a certain characteristic of the entire population. Unlike a sample statistic, which is derived from a subset of the population, the population parameter includes the entire group. Examples of population parameters include the mean, median, variance, and proportion.

Since it is often impractical to measure every individual in a large population, samples are used to estimate these parameters. The confidence interval gives us a likely range for these parameters, allowing us to make educated guesses about the population as a whole. In our context, the population parameters are the true mean values ( \( \mu_1 \text{ and } \mu_2 \)) of the two groups being compared.

The **confidence level** is a crucial aspect of constructing confidence intervals. It represents the degree of certainty we have that the interval truly includes the population parameter. Typically expressed as a percentage, a higher confidence level indicates greater certainty.

• For example, a 90% confidence level suggests that if we were to take 100 different samples and construct a confidence interval for each of them, we expect about 90 of these intervals to contain the true population parameter.
• The common choices for confidence levels are 90%, 95%, and 99%. Higher confidence levels result in wider intervals.

In this exercise, a 90% confidence level means we are fairly confident in the conclusion drawn that \( \mu_1 \) is less than \( \mu_2 \).

**Statistical inference** is the process of using data analysis to deduce properties of an underlying probability distribution. It allows us to make judgments about a population based on sampled data, which is crucial when it's impossible or impractical to assess the entire population.

Key techniques of statistical inference include estimation (where we use samples to estimate population parameters like the mean or variance) and hypothesis testing (where we test an assumption regarding a population parameter).

With statistical inference, we gain insights into what the sample data suggests about the broader population. In our exercise, statistical inference helps conclude — with 90% confidence — that the average outcome in one group differs from another as showcased by the negative difference in means.