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In hypothesis testing, the null hypothesis \((H_0)\) is a statement that assumes there is no effect or no change in the phenomena being tested. In the context of the Carpetland sales team, the null hypothesis asserts that the new compensation plan has not affected the average sales per salesperson. This means that, according to the null hypothesis, the average sales remain at $8,000 per week. Mathematically, we express this as \(H_0: \mu = 8000\).The null hypothesis serves as a baseline or default position that we seek to challenge with our data. It is what we assume to be true until evidence suggests otherwise. Because we propose no change under the null hypothesis, it's crucial for ensuring that any apparent effects observed are genuine and not just the result of random variation.

The alternative hypothesis \((H_1)\) is the statement we aim to provide support for in hypothesis testing. It is contrary to the null hypothesis and suggests that there is a significant effect or change. In the case of the Carpetland sales team's incentive plan, the alternative hypothesis argues that the new plan increases the average sales per salesperson. Here, the mathematical representation is \(H_1: \mu > 8000\).This hypothesis takes on a one-tailed test direction since we are specifically looking for an increase in sales, not just any change. The alternative hypothesis is vital because it embodies the effect or outcome we are testing for. We collect data and perform statistical analyses to assess if there is sufficient evidence to support this hypothesis, potentially leading to a rejection of the null hypothesis.

A Type I error in hypothesis testing occurs when we mistakenly reject the null hypothesis when it is actually true. In the Carpetland example, a Type I error would happen if we conclude that the new compensation plan successfully increases average sales, when, in reality, it does not.

• Consequences: Implementing the compensation plan under this false conclusion would mean the company could incur unnecessary expenses and changes without actual improvement in sales.
• This type of error is also referred to as a "false positive" because it indicates a perceived effect or change that does not exist.

Controlling for Type I errors often involves setting a significance level (usually denoted as \( \alpha \)), which dictates the threshold for rejecting the null hypothesis. A common significance level is 0.05, implying a 5% risk of committing a Type I error.

A Type II error arises in hypothesis testing when we fail to reject the null hypothesis when the alternative hypothesis is true. For Carpetland, this means that even if the new compensation plan does indeed boost average sales, we incorrectly conclude that it does not.

• Consequences: The company misses out on potential revenue gains and continued improvements that the effective compensation plan could bring.
• Such errors are labeled as "false negatives" because they signify a missed detection of a real effect or change.

The probability of committing a Type II error is denoted by \( \beta \). The power of a test, calculated as \(1-\beta\), represents the likelihood of correctly rejecting a false null hypothesis, thus aptly detecting true effects.