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The null hypothesis, often represented as \( H_0 \), is a starting assumption in hypothesis testing. In essence, it is the hypothesis that states there is no effect or no difference. It's a neutral statement, often assumed to be true until evidence suggests otherwise.
In the context of our exercise, the null hypothesis presumes that the true average lateral recumbency time, \( \mu \), for horses is 20 minutes. This is expressed as:

• \( H_0: \mu = 20 \)

Understanding the null hypothesis is crucial because it sets the stage for statistical testing. It's the hypothesis we test directly against the alternative hypothesis. If the collected data provides sufficient evidence against the null hypothesis, we may reject it. If not, we do not reject it and infer that there isn't enough evidence to suggest a difference or effect exists.

The alternative hypothesis, denoted as \( H_a \) or sometimes \( H_1 \), is the statement we aim to support through evidence from our data. It indicates the presence of an effect or a difference compared to the null hypothesis.
For our ketamine study on horses, the alternative hypothesis suggests that the true average lateral recumbency time is less than 20 minutes. This can be expressed as:

• \( H_a: \mu < 20 \)

In hypothesis testing, if the data shows enough evidence to support this claim, the null hypothesis may be rejected in favor of the alternative hypothesis.
The alternative hypothesis is essential because it guides the direction of the statistical test. In this one-sided test, we are specifically looking for evidence to show that the average time is less than 20 minutes.

The significance level, symbolized by \( \alpha \), is a threshold that determines how confident we need to be to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
In this exercise, the significance level is set at \( 0.10 \), or 10%. This means there's a 10% risk of concluding that the average lateral recumbency time is less than 20 minutes when in reality, it isn't.

• A lower \( \alpha \) value indicates more stringent evidence is needed to reject the null hypothesis, reducing the chance of a Type I error.
• A higher \( \alpha \) allows for a higher probability of making this error, but might also detect a true effect more easily if it exists.

Setting a clear \( \alpha \) level at the beginning helps us in deciding later whether the observed data is statistically significant.

The test statistic is a crucial value calculated from the sample data. It is used to decide whether to reject the null hypothesis. For this particular problem, we employ a z-test since the sample size is reasonably large and the population standard deviation is not known.
The formula for the z-test statistic is:

• \[ z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]

where:

• \( \bar{x} \) = sample mean
• \( \mu \) = mean under the null hypothesis
• \( s \) = sample standard deviation
• \( n \) = sample size

Using the values provided (\( \bar{x} = 18.86 \), \( \mu = 20 \), \( s = 8.6 \), \( n = 73 \)), our computed z is approximately -1.136.
The test statistic is then compared against a critical value, determined by the chosen significance level and nature of the test (one-tailed or two-tailed), to reach a conclusion on the hypothesis test.