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In statistics, a **proportion** refers to a part of the whole, expressed as a fraction, percentage, or decimal. It represents the ratio of a subset to the entire population.
For example, if out of 100 students surveyed, 15 walk to school, the proportion is \ \( \frac{15}{100} = 0.15 \ \).

When conducting surveys or experiments, calculating proportions allows us to understand and compare different subsets of data within a population. This is particularly useful in hypothesis testing to determine if observed sample proportions differ from specified values such as claims or expectations.

• Proportions are used to draw conclusions about a population based on a sample.
• They provide a simple way to report the prevalence of a particular characteristic within a group.

The **Null Hypothesis** (often represented as \( H_0 \)) is a fundamental concept in hypothesis testing. It states that there is no effect or no difference, and it serves as the starting point for statistical testing.
For DeAnna's study, the null hypothesis is \( p = 0.13 \), meaning the proportion of students who walk to school is 13\% or less.

The null hypothesis is usually set up to be tested against the alternative hypothesis, \( H_a \), which suggests that the actual effect or difference exists.

Key aspects of \( H_0 \):

• It is assumed true until evidence suggests otherwise.
• Serves as a baseline for comparison to observe sample data.
• Its rejection implies support for the alternative hypothesis.

Understanding the null hypothesis is crucial because it guides the direction and interpretation of statistical tests.

The **Test Statistic** is a value calculated from sample data that is used to evaluate the null hypothesis. It quantifies the degree of agreement between the observed data and \( H_0 \).
In proportion hypothesis testing, the test statistic is often expressed as a \( z \)-score.

For DeAnna's survey, the formula for the test statistic is:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]
where:
\( \hat{p} \) = sample proportion (0.15),
\( p_0 \) = hypothesized population proportion (0.13),
\( n \) = sample size (100).

A higher absolute \( z \)-score indicates that the observed proportion deviate more from the hypothesized proportion.

• The test statistic helps to determine whether to reject \( H_0 \).
• A large \( |z| \)-value suggests that the sample proportion is significantly different from the hypothesized proportion.

The **P-value** measures the probability of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true.
It helps determine the statistical significance of the test.

In hypothesis testing, a small P-value indicates strong evidence against \( H_0 \), prompting researchers to reject the null hypothesis. Conversely, a large P-value suggests there is not enough evidence to refute \( H_0 \).

For instance, in DeAnna's test, if the P-value is 0.08, this means there is an 8\% chance of observing a sample statistic as extreme as the one found, if \( H_0 \) were true.

• P-values help assess the strength of evidence against the null hypothesis.
• Common significance levels for making decisions are 0.05, 0.01, and 0.1.

Thus, P-values play an essential role in deciding whether the results of an hypothesis test are statistically significant.