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In Exercises 4.50 and 4.51 , a randomization distribution is given for a hypothesis test, and shows what values of the sample statistic are likely to occur if the null hypothesis is true. Several possible values are given for a sample statistic. In each case, indicate whether seeing a sample statistic as extreme as the value given is (i) reasonably likely to occur when the null hypothesis is true, (ii) unusual but might occur occasionally when the null hypothesis is true, or (iii) extremely unlikely to ever occur when the null hvpothesis is true. Figure 4.13 (a) shows a randomization distribution for a hypothesis test with \(H_{0}: p=0.30\). Answer the question for these possible sample proportions: (a) \(\hat{p}=0.1\) (b) \(\hat{p}=0.35\) (c) \(\hat{p}=0.6\)

Step by step solution

01

Understanding the Hypothesis

The null hypothesis \(H_{0}: p=0.30\), proposes that the population proportion \(p\) equals 0.30.

02

Comparing \(\hat{p}=0.1\) with the Null Hypothesis

\(p=0.1\) is quite a distance from the null value of \(p=0.30\). Hence, getting a sample proportion as extreme as \(\hat{p}=0.1\) is not reasonably likely to occur when the null hypothesis is true. We can categorize the probability of occurrence as (iii) extremely unlikely to ever occur when the null hypothesis is true.

03

Comparing \(\hat{p}=0.35\) with the Null Hypothesis

Comparing \(p=0.35\) with the null hypothesis, it is closer but still not identical. Thus, it's unusual but might occur occasionally when the null hypothesis is true. We can categorize it as (ii) unusual but might occur occasionally when the null hypothesis is true.

04

Comparing \(\hat{p}=0.6\) with the Null Hypothesis

\(p=0.6\) is very far from the null hypothesis. This is not at all likely considering our null hypothesis. So, it is (iii) extremely unlikely to ever occur when the null hypothesis is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Randomization Distribution

A randomization distribution helps us to visualize how the sample statistics would behave if the null hypothesis is true. Imagine tossing a coin multiple times. The outcomes can vary, but they tend to cluster around a certain point if the coin is fair. That's what randomization distribution does for a hypothesis test.

Here's how it works:

• You assume the null hypothesis is true.
• Generate multiple samples based on this assumption.
• Record the statistics from each sample and create a distribution.

This distribution shows us the likelihood of observing various sample statistics. Values near the center of this distribution mean they are quite likely, while those at the tails or extreme ends are rare.
When a test statistic falls towards the outer edges, it challenges our assumption of the null hypothesis being true.

Null Hypothesis

The null hypothesis, often symbolized as \(H_0\), is a vital part of hypothesis testing. It proposes a specific claim or assumption about a population parameter. In our example, the null hypothesis was \(H_0: p=0.30\).

Here’s the essence of what it involves:

• The null hypothesis assumes no effect or no difference in the context of the study.
• It provides a baseline to help us decide if observed data provides enough evidence to reject it.
• Typically, the null hypothesis is an equality, like \(p = 0.30\).

Rejecting the null hypothesis suggests that the observed data is inconsistent with what we expected under the null hypothesis. Remember, failing to reject it, however, does not prove it true; it simply means there's not enough evidence against it.

Sample Statistic

A sample statistic is a numerical measure based on a sample extracted from a larger population. It's like a snapshot of the whole population but crafted from simpler, smaller data.

In hypothesis testing:

• It is used to estimate the population parameter.
• Common statistics include sample mean \(\overline{x}\), sample proportion \(\hat{p}\), or sample standard deviation \(s\).
• We compare this statistic to what's expected under the null hypothesis.

For example, in the exercise given, we looked at sample proportions such as \(\hat{p} = 0.1\). By comparing this to our null hypothesis \(p = 0.30\), we could investigate if this observed statistic is typical or not under the established assumption.
It's a critical tool in the toolbox of a statistician, used to draw conclusions and make decisions.