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A z-test is a statistical test used to determine whether there is a significant difference between the means of two populations or between the mean of a sample and the mean of a population. It is particularly useful when the sample size is large and the population standard deviation is not known.

In the scenario where Farm Associates wants to test if the mean selling time has increased due to drought conditions, a one-sample z-test is the appropriate choice.

Here's why a z-test is chosen in this scenario:

• The sample size is large (n=100), which satisfies one of the conditions for using a z-test.
• The test compares the sample mean (selling time of 94 days) with a known population mean (90 days).

By calculating the z-value, you can assess how far the sample mean has deviated from the population mean in units of the standard deviation. This helps to make an informed decision about whether the difference is statistically significant.

The significance level, often denoted as \( \alpha \), is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true.

In our exercise with Farm Associates, the significance level is set at 0.10, or 10%. This means there's a 10% risk of concluding that selling times have increased when they haven't.

Choosing a significance level is crucial because it influences the confidence in the results of the test.

Some points to consider about significance levels:

• A lower significance level (like 0.05 or 0.01) means stricter criteria for rejecting the null hypothesis, reducing the risk of a Type I error.
• A higher significance level, like 0.10, is less strict, which is often acceptable in preliminary research or when the consequences of a Type I error are not severe.

Understanding the significance level helps interpret the results of the hypothesis test more accurately.

The null hypothesis, denoted as \( H_0 \), is a statement of no effect or no difference. It provides a baseline that the data will be tested against in statistical hypothesis testing.

For the Farm Associates case, the null hypothesis is that the mean selling time has not changed from the known mean of 90 days, or \( \mu = 90 \).

Here are some characteristics of a null hypothesis:

• It is assumed true until evidence suggests otherwise.
• Failing to reject the null hypothesis simply suggests insufficient evidence against it, not proving it true.
• It is usually expressed with an "=" sign, indicating no effect or change.

Rejecting or failing to reject the null hypothesis depends on comparing the test statistic to a critical value derived from the chosen significance level.

The alternative hypothesis, denoted as \( H_1 \) or sometimes \( H_a \), is a statement that contradicts the null hypothesis. It indicates a new effect or difference that the test aims to support.

In the context of the Farm Associates problem, the alternative hypothesis suggests that the mean selling time is greater than 90 days, or \( \mu > 90 \).

Key points about the alternative hypothesis:

• It represents the research question or what you aim to prove.
• Unlike the null hypothesis, it is often expressed with inequalities (such as \( > \), \( < \), or \( eq \)).
• The choice of a one-tailed or two-tailed test affects the structure of the alternative hypothesis. In this case, a one-tailed test is used, as the interest is in testing if the selling time is greater, not just different.

Evaluating the alternative hypothesis involves determining if the sample data provides enough evidence to reject the null hypothesis in favor of this new claim.