A University of Illinois study on aggressive behavior...

A University of Illinois study on aggressive behavior surveyed a random sample of 558 middle school students. When asked to describe their behavior in the last 30 days, 445 students said their behavior included physical aggression, social ridicule, teasing, name-calling, and issuing threats. This behavior was not defined as bullying in the questionnaire. ${ }^{12}$ Is this evidence that more than three-quarters of middle school students engage in bullying behavior? To find out, Maurice decides to perform a significance test. Unfortunately, he made a few errors along the way. Your job is to spot the mistakes and correct them. $$ \begin{array}{l} H_{0}: p=0.75 \\ H_{a}: \hat{p}>0.797 \end{array} $$ where $p=$ the true mean proportion of middle school students who engaged in bullying. $\bullet\quad$A random sample of 558 middle school students was surveyed. $\bullet\quad$ $558(0.797)=444.73$ is at least $10 .$ $z=\frac{0.75-0.797}{\frac{\sqrt{0.797(0.203)}}{445}}=-2.46 ; P$ -value $=2(0.0069)=0.0138$ The probability that the null hypothesis is true is only 0.0138 , so we reject $H_{0}$. This proves that more than three-quarters of the school engaged in bullying behavior.

A University of Illinois study on aggressive behavior surveyed a random sample of 558 middle school students. When asked to describe their behavior in the last 30 days, 445 students said their behavior included physical aggression, social ridicule, teasing, name-calling, and issuing threats. This behavior was not defined as bullying in the questionnaire. ${ }^{12}$ Is this evidence that more than three-quarters of middle school students engage in bullying behavior? To find out, Maurice decides to perform a significance test. Unfortunately, he made a few errors along the way. Your job is to spot the mistakes and correct them.
$$
\begin{array}{l}
H_{0}: p=0.75 \\
H_{a}: \hat{p}>0.797
\end{array}
$$
where $p=$ the true mean proportion of middle school students who engaged in bullying.
$\bullet\quad$A random sample of 558 middle school students was surveyed.
$\bullet\quad$ $558(0.797)=444.73$ is at least $10 .$
$z=\frac{0.75-0.797}{\frac{\sqrt{0.797(0.203)}}{445}}=-2.46 ; P$ -value $=2(0.0069)=0.0138$
The probability that the null hypothesis is true is only 0.0138 , so we reject $H_{0}$. This proves that more than three-quarters of the school engaged in bullying behavior.

Key Concepts

p-value Interpretation

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one that was observed. It does not indicate the probability that the null hypothesis is true. Correct interpretation of the p-value involves comparing it to a predefined significance level to determine whether to reject H0, rather than treating it as a direct measure of the likelihood of the hypothesis.

Z-Statistic in Proportion Tests

When performing a one-sample proportion test, the z-statistic is computed by subtracting the hypothesized proportion from the observed sample proportion and then dividing by the standard error. This calculation must use the null hypothesis value for p to ensure that the test statistic reflects how far the observed data deviates from what is expected under H0. Misapplication or miscalculation, such as using incorrect values or a wrong denominator, leads to erroneous test results.

Sampling Distribution of Proportions

Central to this testing procedure is the concept that, for large enough samples, the sampling distribution of the sample proportion approximates a normal distribution. This approximation is valid when both np and n(1-p) are sufficiently large (commonly at least 10), which ensures that the subsequent application of the z-test is appropriate. Any violation of these assumptions can undermine the validity of the test results.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents a baseline or default claim about a population parameter, while the alternative hypothesis (Ha) is a statement that contradicts this baseline and reflects the researcher's claim. It is crucial to define these hypotheses correctly by using the population parameter (e.g., the true proportion) rather than a sample statistic. This clear distinction avoids confusion and maintains the integrity of the test.

Standard Error of a Proportion

The standard error provides a measure of the sampling variability of a sample proportion and is calculated by the formula SE = sqrt[p(1-p)/n]. Here, p represents the assumed population proportion (often from H0) and n is the sample size. Using the correct value for p (from the null hypothesis) is essential for accurate computation of the test statistic in a proportion hypothesis test.

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