91Ó°ÊÓ

Hypothesis testing is a method in statistics used to make decisions based on data. When we perform a hypothesis test, we start by establishing two hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis \( H_0 \) represents the status quo or the assertion that any observed difference in a sample is due to chance.
In contrast, the alternative hypothesis \( H_a \) is what we want to prove. For example, in our horse anesthesia study, the null hypothesis is that the true average lateral recumbency time \( \mu \) equals 20 minutes. Meanwhile, the alternative hypothesis suggests that this time is less than 20 minutes.
Hypothesis testing involves comparing these hypotheses using sample data to draw conclusions. This process helps us determine whether our sample data provides enough evidence to reject the null hypothesis.

The critical value in hypothesis testing is a threshold that determines the boundary between regions where the null hypothesis can and cannot be rejected. It's derived from the probability distribution of the test statistic under the null hypothesis.
For the problem at hand, we use a t-distribution since the population standard deviation is not known. The critical value indicates the point beyond which lies an area of significance, typically tailing off into regions of highest improbability under the null hypothesis.
In our exercise, with 72 degrees of freedom and a 0.10 significance level, we find a critical value of approximately \(-1.294\). If the test statistic falls beyond this critical value in the negative direction, the null hypothesis can be rejected, pointing towards a significant finding.

The significance level \( \alpha \) is a probability threshold chosen by the researcher, representing the risk of rejecting the null hypothesis when it is actually true. It's a way to quantify the likelihood of making a type I error, or a false positive.
In the context of our horse anesthesia study, the significance level is set at 0.10, or 10%. This means that we are accepting a 10% chance of wrongly rejecting the null hypothesis when it's true.
The significance level dictates the critical value and helps guide the decision-making process in hypothesis testing. A higher significance level increases the critical region under the null hypothesis distribution, allowing for more room to detect a significant effect.

A test statistic is a standardized value resulting from a sample data analysis, used to decide whether to reject the null hypothesis. It summarizes how far the sample mean deviates from the null hypothesis in units of standard error.
For our one-sample t-test, the formula is \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( \mu_0 \) is the hypothesized population mean, \( s \) is the sample standard deviation, and \( n \) is the sample size.
In our example, we compute \( t \approx -1.14 \), indicating the number of standard errors the sample mean is below the hypothesized population mean. This calculated t-value is then compared to the critical value to decide if the observed effect is statistically significant.