Question

Which of the following are legitimate hypotheses? a. \(p=0.65\) b. \(\hat{p}=0.90\) c. \(\hat{p}=0.10\) d. \(p=0.45\) e. \(p>4.30\)

Short answer

The legitimate hypotheses are: a, \(p=0.65\) and d, \(p=0.45\)

Step-by-step solution

Evaluate each hypothesis

The task here requires us to go through each proposed hypothesis and determine whether it meets all the criteria to qualify as a legitimate hypothesis.

Check for the population parameter

Legitimate hypotheses concern population parameters, not sample statistics. In options b and c, \(\hat{p}\) is used, which is a sample statistic, and thus, these do not constitute legitimate hypotheses.

Consider the numerical value

The numerical value in specific hypotheses must be plausible; for a proportion, it would lie between 0 and 1. Assessing options a, d, and e, we note that e, \(p>4.30\), falls outside this range and is therefore not legitimate.

Conclusion of the suitable hypotheses

Considering the above evaluation, we find that the legitimate hypotheses from the given options are a, \(p=0.65\)and d, \(p=0.45\). These represent population proportions and have values between 0 and 1.

Key concepts

Population Parameter

Understanding the population parameter is at the heart of statistical analysis. In essence, the population parameter is a value that represents a certain characteristic of the entire population. For example, when you see the symbol p in a hypothesis testing problem, it usually denotes the true population proportion, such as the proportion of voters who favor a certain candidate in the entire voting population.

This stands in contrast to sample statistics, which are calculated from a subset of the population - a sample. The confusion between population parameters and sample statistics can lead to incorrectly formulated hypotheses. Remember that a legitimate hypothesis in statistical testing pertains to population parameters, not to sample statistics.

Sample Statistics

Flip the coin, and we have sample statistics, which are estimates derived from analyzing a subset of the population — a sample. Sample statistics, such as \(\hat{p}\) which represents the sample proportion, serve as our best guess about the population parameter when examining data from a sample.

The distinction between a population parameter (such as \(p\)) and a sample statistic (like \(\hat{p}\)) is crucial for hypothesis testing. Validating hypotheses involving sample statistics, like ‘b’ and ‘c’ in the provided exercise, would be a misunderstanding of concepts, as hypotheses should directly pertain to population parameters.

Proportion

Proportion, often encountered in statistics, refers to the fraction of the population that features a certain attribute. In formal hypothesis testing, the notion of a proportion typically relates to the population proportion, symbolized as \(p\). A proportion is always a value between 0 and 1, as it represents a part of the whole.

Be alert to impossible proportion values, as seen in the example 'e' from the exercise, where \(p>4.30\) doesn't fall within the range of 0 to 1. Thus, this cannot be considered a legitimate hypothesis. Recognizing valid numerical values for proportions is essential for framing and testing a statistical hypothesis.

Statistical Hypothesis

A statistical hypothesis is the assumption we aim to test through statistical analysis. It posits an initial claim about a population parameter, like a mean or a proportion. There are usually two complementary hypotheses in a test: the null hypothesis (\(H_0\)), which claims that any effect or difference is due to chance, and the alternative hypothesis (\(H_1\) or \(H_a\)), which states that there is indeed a significant effect or difference.

When formulating a statistical hypothesis, it's critical that it is properly structured. It must concern a population parameter, the values need to be plausible (e.g., a proportion that is between 0 and 1), and it should be written in a testable form. For instance, hypotheses 'a' and 'd' from the exercise (\(p=0.65\) and \(p=0.45\)) meet these criteria, thereby qualifying as legitimate.