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The z-test is a statistical method used to determine if there's a significant difference between sample data and a known population mean. It is particularly useful when the population standard deviation is known, which is the case in our example. Here, we use the z-test to test hypotheses about the average of a specific set of observations.
In general, the steps for conducting a z-test are straightforward:

• Compute the z-statistic using the formula: \[ Z_{obs} = \frac{(\overline{x} - \mu)}{(\sigma/ \sqrt{n})} \]
• Compare this observed z-value against a critical value that corresponds to your chosen significance level (often denoted as \( \alpha \)).

The calculated z-value tells us how many standard deviations our sample mean is from the population mean under the null hypothesis. In hypothesis testing, the z-test helps us to either reject or fail to reject the null hypothesis based on this figure.

The significance level, represented as \( \alpha \), indicates the probability of rejecting the null hypothesis when it is actually true. It is a crucial component that helps to define our acceptance criteria for the hypothesis test.
In most hypothesis tests, a typical \( \alpha \) level is 0.05, meaning we have a 5% risk of concluding that there is an effect or difference when there is none. This level is often known as the level of risk, and helps to handle the concept of Type I error.
Choosing a significance level is a balance between risk and reliability. A smaller \( \alpha \) suggests higher confidence, but decreases the sensitivity to detect an actual effect, while a larger \( \alpha \) does the opposite.

The critical value is a threshold which the test statistic must exceed in order to reject the null hypothesis. It is determined based on the chosen significance level and the nature of the test (e.g., two-tailed or one-tailed).

In our problem, the critical z-value for the two-tailed test is approximately ±1.96, and for the one-tailed test, it is 1.645. These values can be found in standard z-tables. The decision rule is simple:

• If our observed z-value surpasses the critical threshold (in magnitude), we reject the null hypothesis.
• If it does not, then we cannot reject the null hypothesis.

This logic helps understand whether the sample mean indicates a significant difference from the hypothesized population mean.

A two-tailed test is used when changes in either direction away from the hypothesized parameter are considered significant. This type of test will check for both positive and negative differences.

In our example with the hypothesis \( H_{1}: \mu eq 8.5 \), it implies that the population mean could be either less than or greater than 8.5.
The critical z-value for a significance level of 0.05 is ±1.96. If the absolute value of our observed z-value is greater than this, we reject the null hypothesis.
Two-tailed tests are common in scientific research where deviation in either direction is of interest.

In a one-tailed test, the hypothesis specifically targets a deviation in a single direction. This is appropriate when the direction of the anticipated effect is known and confidently one-sided.

Considering our hypothesis test of \( H_{1}: \mu > 8.5 \), we exclusively investigate if the sample mean is significantly greater than the hypothesized mean. For such a test with a significance level of 0.05, the critical value of z translates to approximately 1.645.

If our observed z-value is greater than this critical value, we conclude by rejecting the null hypothesis. A one-tailed test thus is precise and direct, making it efficient when the direction of interest is clear and one-sided.