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The z-score is a statistical figure that indicates how many standard deviations an element is from the mean. In hypothesis testing, it helps determine if the observed data significantly deviates from the null hypothesis.
To calculate the z-score, we use the formula: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]where:

• \(\bar{x}\) is the sample mean
• \(\mu\) is the population mean under the null hypothesis
• \(\sigma\) is the population standard deviation
• \(n\) is the sample size

By plugging these values into the formula, we get a z-score that helps determine whether the observed mean is significantly different from the hypothesized mean. If the z-score is far from zero, it indicates a higher degree of deviation from the null hypothesis mean.
In our examples, we compute the z-score for each hypothesis to see how the sample mean compares to what is expected under the null hypothesis.

The critical value is an essential part of hypothesis testing. It marks the boundary or cutoff point beyond which the null hypothesis will be rejected.
Critical values differ depending on the type of test being conducted (two-tailed, left-tailed, or right-tailed) and the significance level \(\alpha\).

• In a two-tailed test, critical values form a range, with both positive and negative cutoffs.
• In left or right-tailed tests, there is only one critical value to be considered.

For the significance level \(\alpha\), we determine how confident we want to be about our decision to reject or accept the null hypothesis.
For example, in our exercise:

• With a two-tailed test at \(\alpha = 0.10\), the critical values are \(-1.645\) and \(+1.645\).
• In a left-tailed test at \(\alpha = 0.01\), the critical value is \(-2.33\).
• In a right-tailed test at \(\alpha = 0.05\), the critical value is \(+1.645\).

Comparing the calculated z-score with these critical values guides the decision on whether to reject the null hypothesis.

The null hypothesis \(H_0\) is a default statement that assumes no effect or no difference in the population parameter being tested. It is the hypothesis that researchers typically aim to challenge or test against.
\(H_0\) generally posits that there is no change, no difference, or no effect. In our exercise examples, the null hypotheses suggest certain population means, such as \(\mu = 80\), \(\mu = 32\), and \(\mu = 55\).
In the context of hypothesis testing, we either retain the null hypothesis when the data supports it, or reject it when sufficient evidence suggests an alternative scenario. The null hypothesis serves as a critical baseline for statistical testing.

The alternative hypothesis \(H_1\) explains what the null hypothesis does not. It suggests that there is a significant effect, difference, or change.
\(H_1\) is what researchers desire to prove. It indicates the presence of conditions such as a mean different from a certain value, a mean less than a particular number, or a mean greater than a particular number.

• In our two-tailed test example, \(H_1: \mu eq 80\) indicates mean different from 80.
• For the left-tailed test, \(H_1: \mu < 32\) anticipates a mean less than 32.
• In the right-tailed test scenario, \(H_1: \mu > 55\) hypothesizes a mean greater than 55.

The role of the alternative hypothesis is pivotal, as it guides decision-making in statistical testing. Rejecting the null hypothesis in favor of the alternative influences conclusions drawn from the analysis.