91Ó°ÊÓ

When dealing with two population means, the main goal is often to compare them to determine if there is a significant difference. In this context, population means refer to the average values of a particular variable in two distinct groups. For example, if we have two groups with measurements,

• Population 1 could represent the mean value \( \mu_1 \) , and
• Population 2 could represent the mean value \( \mu_2 \) .

We make use of data collected from samples to estimate these means and use statistical methods to test any claims about their differences. In statistical testing, you often check whether the means are equal, one is greater than the other, or simply not equal. The null hypothesis usually starts with the assumption that the means are equal. In this exercise's context, we are testing the claim concerning the difference between \( \mu_1 \) and \( \mu_2 \). When the sample means are different, you'll compute a test statistic to better understand whether this difference is statistically significant or possibly due to random sampling variation.

The level of significance, denoted by \( \alpha \), is a threshold set by the researcher that determines how much risk of a Type I error, or false positive, is acceptable. In simpler terms, it's the probability with which we are willing to incorrectly reject the null hypothesis.
For instance, a typical level of significance might be \(0.05\), which indicates a 5% risk. This means that if the null hypothesis were true, there would be a 5% chance of observing results at least as extreme as those obtained.
In hypothesis testing, you compare the p-value of your test (not directly mentioned here) to this \(\alpha\) to decide if your results are significant. A smaller level of significance means stricter criteria for concluding that the results are significant. This is crucial because it balances the trade-off between Type I and Type II errors, helping ensure that your results are reliable and valid.

The critical value is a point on the test statistic distribution that represents the threshold to which the test statistic is compared. It plays a key role in hypothesis testing.
For example, if we have a left-tailed test, we determine the critical value that corresponds to our chosen level of significance, \(\alpha\). In this scenario, if \(\alpha = 0.05\), we use a z-table to find the critical z-value, which is approximately \(-1.645\).
The critical value allows you to determine the rejection region in the distribution. Depending on the type of test (one-tailed or two-tailed), the rejection region will lie on one side or both sides of the distribution.

• If the test statistic falls in this rejection region, you reject the null hypothesis.
• If it doesn’t, as seen in our exercise where the calculated z-statistic was greater than the critical value, you fail to reject the null hypothesis.

In hypothesis testing, the test statistic is a standardized value you calculate from sample data. It's a crucial step that helps you make a decision about the hypotheses.
In our exercise, the test statistic was calculated using the formula:
\( z = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\).
This formula compares the difference between the sample means \( \bar{x}_1 \) and \( \bar{x}_2 \) to zero, dividing by the standard error of their difference.

• \( \bar{x}_1 \) and \( \bar{x}_2 \) are the respective sample means.
• \( \sigma_1 \) and \( \sigma_2 \) are the population standard deviations.
• \( n_1 \) and \( n_2 \) are the sample sizes.

The resulting value, in our case \(-1.429\), is compared against the critical value to decide whether we reject the null hypothesis or not.