Question

Determine whether the sets of hypotheses given are valid hypotheses. State whether each set of hypotheses is valid for a statistical test. If not valid, explain why not. (a) \(H_{0}: \mu=15 \quad\) vs \(\quad H_{a}: \mu \neq 15\) (b) \(H_{0}: p \neq 0.5 \quad\) vs \(\quad H_{a}: p=0.5\) (c) \(H_{0}: p_{1}p_{2}\) (d) \(H_{0}: \bar{x}_{1}=\bar{x}_{2} \quad\) vs \(\quad H_{a}: \bar{x}_{1} \neq \bar{x}_{2}\)

Short answer

(a) Valid; (b) Not valid, not comprehensive; (c) Not valid, not comprehensive; (d) Valid

Step-by-step solution

Evaluate Hypotheses in (a)

In (a), the null hypothesis \(H_{0}: \mu=15\) suggests that the population mean \(\mu\) equals 15. The alternative hypothesis \(H_{a}: \mu \neq 15\) suggests the mean \(\mu\) is not 15. These hypotheses are mutually exclusive and comprehensive, hence, this is a valid set of hypotheses.

Evaluate Hypotheses in (b)

In (b), the null hypothesis \(H_{0}: p \neq 0.5\) suggests that the population proportion \(p\) is not 0.5. The alternative hypothesis \(H_{a}: p=0.5\) suggests that \(p\) equals 0.5. These hypotheses are mutually exclusive but they are not comprehensive, because they don’t cover the possibility of \(p\) being exactly 0.5. Therefore, this is not a valid set of hypotheses.

Evaluate Hypotheses in (c)

In (c), the null hypothesis \(H_{0}: p_{1}p_{2}\) suggests that \(p_{1}\) is greater than \(p_{2}\). These hypotheses are not comprehensive as they don't cover the possibility of \(p_{1}\) being equal to \(p_{2}\). Therefore, this is not a valid set of hypotheses.

Evaluate Hypotheses in (d)

In (d), the null hypothesis \(H_{0}: \bar{x}_{1}=\bar{x}_{2}\) asserts that the sample means \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are equal. The alternative hypothesis \(H_{a}: \bar{x}_{1} \neq \bar{x}_{2}\) suggests they are not equal. These hypotheses are mutually exclusive and comprehensive. Hence, these constitute a valid set of hypotheses.

Key concepts

Null Hypothesis

The null hypothesis, denoted as H₀, is a statement in statistical hypothesis testing that there is no effect or no difference, and it serves as a starting point for testing. It posits that any observed effect or difference is due to chance. For example, if we are testing a new teaching method, the null hypothesis would state that the method has no impact on student performance. In the given exercise (a), H₀: \(\mu=15\) implies that the population mean is believed to be 15 before any testing is done. In hypothesis testing, we collect data to assess the plausibility of the null hypothesis. If evidence suggests that the null hypothesis is likely untrue, it may be rejected in favor of the alternative hypothesis.

Alternative Hypothesis

Conversely, the alternative hypothesis, represented as H_a, posits that there is an effect or a difference, and it is directly contrary to the null hypothesis. This hypothesis is what researchers aim to support. In our teaching method example, the alternative hypothesis would suggest that the new method does improve student performance. The H_a in exercise (a), H_a: \(\mu eq 15\), indicates the belief that the population mean is not 15. The alternative hypothesis is only accepted when there is sufficient evidence to prove that the null hypothesis is untenable beyond a reasonable doubt.

Hypothesis Testing

Hypothesis testing is a systematic process used to evaluate the plausibility of a hypothesis using sample data. It involves comparing the observed data to what we would expect to see if the null hypothesis were true. If the data departs significantly from the null hypothesis, it suggests the alternative hypothesis may be true. The decision to accept or reject the null hypothesis is based on the test statistic and its associated p-value or on a confidence interval. If the data collected during the experiment falls into the region of rejection, defined before the test, then the null hypothesis is rejected in favor of the alternative hypothesis.

Population Mean

The population mean, represented by the Greek letter \(\mu\), is the average value in a population. It is a vital concept in statistics as it provides a central location of the data. When we can't measure every individual in a population, we estimate the mean using a representative sample. This value becomes a prediction of the population mean. For example, in exercise (a), the population mean is specifically addressed in the null hypothesis H₀: \(\mu=15\), suggesting that we expect the true population mean to be 15.

Population Proportion

Population proportion, denoted as p, is the ratio of individuals in a population that possess a certain attribute or characteristic. For instance, it could be the proportion of people who prefer online learning over traditional classrooms. It's the target value we often want to estimate or make inferences about. In exercise (b), H_a: p=0.5 refers to the hypothesis that exactly half of the population possesses the characteristic in question. It's crucial to remember that not all hypotheses about proportions are valid, as seen in cases (b) and (c) which were invalidated due to comprehensiveness issues.

Sample Means

Sample means, indicated by \(\bar{x}\), are the averages calculated from samples, which are subsets taken from the population. They are used as estimates of the population mean. Because it's impractical to study an entire population, a sample provides a manageable and cost-effective snippet of the total group. The accuracy of the sample mean as an estimate of the population mean depends on sample size and variance within the data. In exercise (d), the hypothesis H₀: \(\bar{x}_{1}=\bar{x}_{2}\) is testing whether the mean of one sample is equal to the mean of another, an essential consideration in comparing two groups.