91Ó°ÊÓ

Statistical significance is a fundamental concept in hypothesis testing. It measures the probability that a given result or relationship is due to something other than mere chance. In the context of hypothesis testing, when a result is statistically significant, it signals that we have enough evidence to reject the null hypothesis.

In our example, the null hypothesis (\(H_0: \mu = 0\)) is compared against an alternative hypothesis (\(H_a: \mu > 0\)). A test is considered statistically significant if the calculated test statistic (like the z-statistic) is greater than a pre-determined value called the critical value.

Statistical significance is always assessed in terms of a specific probability level, known as the significance level, typically indicated by \( \alpha \). If a result is significant at a \(5\%\) level, that means there's a \(5\%\) chance the result is due to random variation alone.

The critical value is a specific threshold that distinguishes whether the test statistic indicates a statistically significant result. In hypothesis testing, critical values are determined based on the desired significance level \((\alpha)\) and the type of test (one-tailed or two-tailed).

For a one-tailed test at a \(5\%\) significance level, the critical z-value is usually about \(1.645\). This indicates that any test statistic higher than this value shows evidence strong enough to reject the null hypothesis. While at a \(1\%\) significance level, the critical z-value increases to around \(2.33\).

To determine critical values, you can consult a z-table, which lists z-values corresponding to various significance levels and tail types. By setting appropriate critical values, researchers can effectively control the rate of type I errors, or false positives, whereby the null hypothesis is wrongly rejected.

The z-statistic is a measure that determines how far away a sample mean is from the null hypothesis population mean, in terms of standard deviations. It is calculated during hypothesis testing when the population standard deviation is known.

In our exercise, the z-statistic is given as \(z = 1.65\). This value provides a way to compare sample data against a population under the null hypothesis. If the z-statistic is greater than the critical value, this indicates that the result is significant, and the likelihood that the observed data is due to random chance is low.

The z-statistic is critical for converting the raw difference between observed and expected values into a standard score, making it easier to infer significance. This conversion involves using the z-score formula, which calculates how many standard deviations an element is from the mean, facilitating comparisons across different data contexts.

Significance levels, denoted as \(\alpha\), are predetermined probability thresholds that define how likely a result would be due to random chance. Commonly used levels are \(0.05\) (\(5\%\)) and \(0.01\) (\(1\%\)), with lower alpha levels signaling stricter criteria for claiming statistical significance.

These significance levels play a vital role in hypothesis testing because they set the standard for rejecting or failing to reject the null hypothesis. A result is deemed significant if the calculated p-value is less than \(\alpha\). For instance, if you set \(\alpha = 0.05\), you are accepting a \(5\%\) risk of concluding that a difference exists when there is none (type I error).

Choosing appropriate significance levels involves balancing the risk of false positives with the need for sensitivity in detecting true effects. Researchers often decide on \(\alpha\) based on the discipline or context of the study, where more stringent fields might prefer a \(0.01\) level, stressing the importance of robust evidence before claiming an effect.