For each of the following assertions, state whether it is a legitimate statistical hypothesis an

For each of the following assertions, state whether it is a...

Key Concepts

Statistical Hypothesis

A statistical hypothesis is a formal statement about a population parameter (or a set of parameters) that can be tested using statistical evidence. It must be stated in terms of parameters defined within a specified model and be contextually verifiable using data and probability models.

Population Parameter

A population parameter is a numerical summary measure that describes an aspect of a population, such as a mean, variance, or proportion. Hypotheses must refer to these parameters rather than to sample-derived statistics, ensuring that the statement concerns the underlying distribution rather than just the observed data.

Differentiating Sample Statistics from Parameters

In hypothesis testing, it is crucial to distinguish between sample statistics (which are computed from observed data, like a sample mean or sample proportion) and the true population parameters. Only parameters can be the subject of statistical hypotheses because they are the quantities of interest that models aim to describe and estimate.

Testability and Proper Formulation

For a hypothesis to be legitimate, it needs to be testable. This means it is formulated in terms of clearly defined parameters within a statistical model, and there exists a procedure whereby the hypothesis can be evaluated against data through inference. Hypotheses that make claims about sample outcomes or estimates without referencing the population parameter do not meet this requirement.

Use of Equality and Inequality in Hypotheses

Hypothesis statements may be constructed using equalities or inequalities, such as asserting that a parameter equals a specific value or lies within a certain range. However, these statements must always pertain to the underlying population parameter to be a valid hypothesis, as using sample estimates or summary measures directly undermines the generalization intended by the hypothesis.

Transcript

00:01 Okay, so to answer this question, we have to ask ourselves, when is a statistical hypothesis legitimate? okay, so a statistical hypothesis is legitimate if it is a statement about the population parameter.

00:22 And you might want to think of statistical hypothesis as illegitimate if it is a statement.

00:31 Making a statement about the sample statistic.

00:34 So if it's a population parameter, that's correct.

00:38 And if it's a sample statistic, that's wrong.

00:43 Okay, so looking at the problem that we have here, in part a, the hypothesis is about sigma being greater than 100.

00:54 So this is a legitimate statistical hypothesis.

01:01 Because sigma is a symbol which we use to denote the population standard deviation.

01:11 So this is a legitimate statistical hypothesis.

01:14 And the second hypothesis, which is in part b, is about phead being equal to 0 .45.

01:24 And this is not a legitimate statistical hypothesis, because it is a legitimate statistical hypothesis.

01:31 Because it is making a statement about p head, which we denote for the sample proportion.

01:39 So this is not a legitimate statistical hypothesis.

01:46 Okay, part c, we have s being less than or equal to 0 .2.

01:54 And this is not a legitimate statistical hypothesis because, it's making a statement about s which we use to denote the sample standard deviation.

02:09 All right.

02:09 So this is an incorrect hypothesis.

02:14 Okay, on part d, we have sigma 1 divided by sigma 2 being less than 1.

02:24 So sigma 1 is a symbol which we used to denote the population standard deviation for population 1 and sigma 2 is, you know, again, the symbol which we used to denote the population and standard deviation for population 2.

02:44 And so the ratio of this two is still a ratio calculated for the population parameters...

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