91Ó°ÊÓ

In hypothesis testing, the null hypothesis is your starting assumption. It's like saying, "There is no change." For the real estate agency in Nebraska, the null hypothesis (\( H_0 \)) assumes that the average selling time of farm properties remains at 90 days. This is a baseline or default position. It asserts that the recent drought has not affected the selling time significantly.

Writing the null hypothesis mathematically helps clarify this point: \( H_0: \mu = 90 \), where \( \mu \) is the mean selling time. Notice how straightforward this is? You're essentially trying to see if there's enough evidence to move away from this assumption, to show that the selling time has actually changed due to conditions like drought.

Keeping it simple, the null hypothesis acts as the claim you're testing against. You're looking for evidence strong enough to reject this claim.

The alternative hypothesis steps onto the stage when you have reasonable evidence that the initial assumption (null hypothesis) is outdated. For the situation with Farm Associates, the alternative hypothesis (\( H_a \)) posits that the average selling time is more than 90 days. This is directly derived from the agency's suspicion about increased selling times.

In mathematical terms, it's expressed as: \( H_a: \mu > 90 \). The alternative hypothesis provides a specific direction for change, suggesting that the drought conditions have indeed affected the selling time.

Think of the alternative hypothesis as where the action happens - it's the new theory you're testing for. You'd accept it only if you have enough statistical evidence to back it up. Here, we're interested in seeing if the selling time is indeed longer than before, which would have practical implications for the real estate agency in question.

The t-test is a common statistical method used to determine if there's a significant difference between a sample mean and a known population mean. It's particularly useful when the population standard deviation is unknown, and the sample size isn't too small, making it ideal for the Farm Associates exercise with 100 samples.

Here's a simple breakdown of how the t-test works for this scenario:

• Calculate the t-test statistic using the given formula: \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \), where \( \bar{x} = 94 \), \( \mu_0 = 90 \), \( s = 22 \), and \( n = 100 \).
• Plug in the numbers: \( t = \frac{94 - 90}{22/\sqrt{100}} \approx 1.818 \).
• Compare this computed t-value to a critical value from the t-distribution table at a \(0.10d\) significance level. If the t-value is greater, you reject the null hypothesis.

The t-test essentially helps in making a decision about the null hypothesis. It provides the evidence necessary to either stick with or reject the initial assumption. In this exercise, since the computed t-value is greater than the critical value, there is enough evidence to conclude that the drought indeed increased the average selling time.