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When you want to know if the average of your sample is different from a known population mean, and you already know the population's standard deviation, the z-test is a common go-to. It's often used in situations where you have a large enough sample size - typically over 30 - which makes the normal distribution assumption reasonable. The form of the test statistic

• Given by the formula: \[ Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]
• Where \( \bar{x} \) is your sample mean, \( \mu \) is the known population mean, \( \sigma \) is the population standard deviation, and \( n \) is your sample size.

Let's break it down: if you have a sample with an average value, and you're curious about how far off it is from what you'd expect from the population, the z-test helps you quantify that difference. In simple terms, you're checking how many standard deviations away your sample mean is from the population mean. The further away it is, the more likely there's something going on with your sample.

In the exercise you looked at, we calculated a z-score for different sample means. For instance, a sample mean of 23 resulted in a z-score of 0.866. This helps us understand how typical or unusual these average results are under the hypothesis.

The p-value is an important part of hypothesis testing. This number tells you how probable your observation would be if the null hypothesis is true. The smaller the p-value, the more evidence you have against the null hypothesis. In other words, it's a handy way to judge the strength of your evidence!

• If your p-value is less than or equal to your significance level (often denoted by \( \alpha \)), you reject the null hypothesis.
• In the textbook example, we used \( \alpha = 0.01 \).

For example, if you obtained a sample mean of 25.1, the p-value was 0.0072. This is smaller than 0.01, suggesting strong evidence against the null hypothesis. Conversely, a sample mean of 23 resulted in a higher p-value of 0.3872, indicating weak evidence against it.

By calculating the p-value, researchers can decide whether their sample result is just a random fluke or it's really indicative of a different population mean. The p-value thus acts like a thermometer, telling you how 'hot' (or unusual) your sample data really is compared to the null expectation.

A two-tailed test is all about checking if there are differences in either direction. This test is suitable when you're open to finding results that are either significantly higher or lower than the population mean.

• The two-tailed test considers deviations on both sides of the distribution tail.
• This means you must consider the extremities on both ends for possible outcomes.

In the exercise provided, the two-tailed nature was indicated by testing whether \( \mu eq 22 \). Under this hypothesis, you're not just concerned if the average is larger or smaller than 22, but rather if it just isn't 22.

With a two-tailed approach, you double the p-value of one tail test. For instance, if a z-score finds that the right tail probability is 0.1936, the two-tailed p-value becomes 0.3872 (as it incorporates both tails). Consider this like weighing all your options equally, whether they're above or below the assumed average.