Question

For the following pairs, indicate which do not comply with the rules for setting up hypotheses, and explain why: a. \(H_{0}: \mu=15, H_{a}: \mu=15\) b. \(H_{0}: \pi=.4, H_{a}: \pi>.6\) c. \(H_{0}: \mu=123, H_{a}: \mu<123\) d. \(H_{0}: \mu=123, H_{a}: \mu=125\) e. \(H_{0}: p=.1, H_{a}: p=125\)

Short answer

The pairs which do not comply with the rules for setting up hypotheses are a. \(H_{0}: \mu=15, H_{a}: \mu=15\), b. \(H_{0}: \pi=.4, H_{a}: \pi>.6\), d. \(H_{0}: \mu=123, H_{a}: \mu=125\) and e. \(H_{0}: p=.1, H_{a}: p=125\). The reasons include - both hypotheses being the same (pair a), not covering all possible values of the parameter (pair b), hypotheses not being mutually exclusive (pair d), and incorrect parameter value (pair e).

Step-by-step solution

Analyzing first pair

In pair a. \(H_{0}: \mu=15, H_{a}: \mu=15\), both hypotheses are the same. This violates the rule that \(H_{0}\) and \(H_{a}\) are mutually exclusive. Hence, this pair doesn't comply with the rules.

Analyzing second pair

In pair b. \(H_{0}: \pi=.4, H_{a}: \pi>.6\), the sum of \(H_{0}\) and \(H_{a}\) doesn’t cover values between .4 and .6 which violates one of the rules, hence, this pair also doesn't comply.

Analyzing third pair

In pair c. \(H_{0}: \mu=123, H_{a}: \mu<123\), the hypotheses are mutually exclusive and the equality sign is in the correct hypothesis. So, this pair complies with the rules.

Analyzing fourth pair

In pair d. \(H_{0}: \mu=123, H_{a}: \mu=125\), we see that these two hypotheses are not mutually exclusive which violates one of the rules. Hence, this pair doesn't comply with the rules.

Analyzing fifth pair

In pair e. \(H_{0}: p=.1, H_{a}: p=125\), the alternative hypothesis clearly doesn’t make sense because a probability should lie between 0 and 1 inclusive, hence this pair does not comply with the rules.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is fundamental in hypothesis testing, an integral part of statistics. The null hypothesis, denoted as \(H_{0}\), typically represents a statement of no effect or no difference. It is the hypothesis that researchers aim to test against the alternative hypothesis. The alternative hypothesis, expressed as \(H_{a}\), proposes what we suspect might actually be true or what we wish to test for. For example, if we want to test if a new drug is more effective than the current standard, the null hypothesis would say the new drug has the same effectiveness as the current one, while the alternative would claim it is more effective.

The two hypotheses must be framed so they are mutually exclusive and exhaustive; that means they cannot overlap and must cover all possible outcomes. An error in setting these can invalidate the entire test. For instance, if the null says a coin is fair (\(p = 0.5\)), the alternative should be that the coin is not fair (\(p eq 0.5\)), thus covering all other possibilities, such as \(p > 0.5\) or \(p < 0.5\). It's essential to state these hypotheses correctly and clearly as they set the stage for all subsequent steps in hypothesis testing.

Mutually Exclusive Hypotheses

The concept of 'mutually exclusive hypotheses' can sometimes be a tricky one for students. Essentially, hypotheses are mutually exclusive if they do not overlap and cannot both be true at the same time. In any hypothesis test, the null and alternative hypotheses should be mutually exclusive to ensure a clear decision can be made from the test results. In the exercise provided, the fourth pair \(H_{0}: \mu=123, H_{a}: \mu=125\) is an example of hypotheses that are not mutually exclusive. This is incorrect because they do not cover all possibilities and both could potentially be true at the same time. If the population mean \(\mu\) was actually 124, both \(H_{0}\) and \(H_{a}\) would be incorrect. Instead, the alternative hypothesis should be something like \(H_{a}: \mu eq 123\) to contrast \(H_{0}\) and cover all other values except 123.

When hypotheses are mutually exclusive, only one can be true. This is vital for the validity of the test, for interpreting results correctly, and for making sound conclusions. If this criterion isn't met, the entire hypothesis testing process can lead to ambiguous or misleading outcomes, rendering the analysis scientifically weak or useless.

Probability in Statistics

Probability plays a central role in statistics, and it forms the backbone of hypothesis testing. It is the measure of the likelihood that an event will occur, expressed as a number between 0 (the event will not occur) and 1 (the event will occur for sure). For instance, when a coin is flipped, the probability of landing a head or a tail is each \(0.5\) if the coin is fair. This is because there are only two equally likely outcomes.

In hypothesis testing, we use probability to make conclusions about the population based on sample data. We consider the probability of observing our sample data, or something more extreme, if the null hypothesis were true. This p-value helps us decide whether to reject or fail to reject the null hypothesis. If the p-value is below a certain threshold (commonly \(0.05\)), we reject the null hypothesis in favor of the alternative.

The fifth pair in the exercise, \(H_{0}: p=.1, H_{a}: p=125\), misunderstands the concept of probability since probabilities cannot exceed 1. A correct alternative hypothesis might state \(H_{a}: p eq .1\) or \(H_{a}: p > .1\) depending on the context, reflecting probabilities that are consistent with defined boundaries (0 to 1). Grasping the nature of probability is crucial because it helps us quantify uncertainty and make informed decisions based on statistical evidence.