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The Z-statistic is a measure used in hypothesis testing when dealing with population samples. It tells us how far, in standard deviations, our sample mean is from the population mean under the null hypothesis. This statistic is particularly useful when the sample size is large (typically over 30), as the Central Limit Theorem assures us the distribution of the sample mean will be approximately normal.

The formula for calculating the Z-statistic is given by:\[Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\]where:

• \(\bar{x}\) is the sample mean,
• \(\mu\) is the population mean,
• \(\sigma\) is the population standard deviation,
• \(n\) is the sample size.

Once the Z-statistic is calculated, it is compared against critical values from the standard normal distribution to determine statistically significant differences between the sample and population means.

The significance level, denoted by \(\alpha\), is the threshold we set to decide whether our statistical test results are meaningful. Common significance levels used in hypothesis testing are 0.05 and 0.01. Essentially, a significance level is the probability of rejecting the null hypothesis when it is actually true, also known as the "Type I error rate."

Choosing a lower \(\alpha\) (e.g., 0.01 instead of 0.05) means we have stricter criteria for claiming a significant result, reducing the risk of a Type I error. However, this also increases the chance of a Type II error, where the test fails to detect a true effect. In the context of the given exercise, the test was significant at \(\alpha=0.05\) because the Z-statistic exceeded the critical value of 1.645. However, it was not significant at \(\alpha=0.01\), since the Z-statistic did not exceed 2.33, the higher threshold for significance.

The null hypothesis, abbreviated as \(H_0\), forms a critical part of hypothesis testing. It is the starting assumption that there is no effect or difference, and any observed variation is due to random chance. In statistical terms, the null hypothesis often posits that there is no association between variables, or no difference between groups.

The main goal of hypothesis testing is to determine whether there is enough evidence in the sample data to reject \(H_0\). If the evidence is strong enough, indicated by the test statistic and significance level, we reject the null hypothesis. Otherwise, we fail to reject it. Importantly, failing to reject \(H_0\) does not prove it is true; it merely indicates insufficient evidence against it. In our example, the preset threshold for the test was exceeded at \(\alpha=0.05\), suggesting enough evidence to reject \(H_0\) at this level.

The p-value is a measure of the probability that the observed data would occur by random chance under the null hypothesis. It is calculated during hypothesis testing and serves as a decision tool for determining statistical significance.

If the p-value is less than the chosen significance level, \(\alpha\), we have strong evidence to reject the null hypothesis. This indicates the observed data are unlikely due to random variation alone. A small p-value suggests that the null hypothesis is less likely to be true.

For the exercise provided, the p-value associated with the Z-statistic of 2.29 would be calculated and compared against the significance levels (\(\alpha=0.05\) and \(\alpha=0.01\)). Since the Z-statistic was significant at 0.05 but not at 0.01, the corresponding p-value would be smaller than 0.05 but larger than 0.01, indicating a result that is statistically significant at the 0.05 level.