Bullies in middle school A University of Illinois study on...

Bullies in middle school 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. Is this evidence that more than three-quarters of the students at that middle school 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. A random sample of 558 middle school students was surveyed. $558(0.797)=444.73$ is at least 10 $$ z=\frac{0.75-0.797}{\sqrt{\frac{0.797(0.203)}{445}}}=-2.46 ; P \text { -value }=2(0.0069)=0.0138 $$ 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.

Bullies in middle school 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. Is this evidence that
more than three-quarters of the students at that middle school 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.
A random sample of 558 middle school students was surveyed.
$558(0.797)=444.73$ is at least 10
$$
z=\frac{0.75-0.797}{\sqrt{\frac{0.797(0.203)}{445}}}=-2.46 ; P \text { -value }=2(0.0069)=0.0138
$$
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

Interpretation of the p-value

The p-value is defined as the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. It is not the probability that the null hypothesis itself is true. Misinterpreting the p-value as the chance that the null hypothesis is correct leads to invalid conclusions about the data. Correct interpretation is crucial for making sound decisions regarding H0.

Test Statistic Calculation

The test statistic in a significance test for proportions is computed by taking the difference between the sample proportion and the hypothesized proportion (from H0) and then dividing by the standard error calculated under H0. This z-score measures how many standard deviations the sample estimate is away from the null hypothesis value. A mistake in either the numerator (by using an incorrect alternative value) or the denominator (by miscalculation of the standard error) results in an incorrect test statistic and misleading p-value.

One-tailed vs. Two-tailed Tests

The selection between a one-tailed and a two-tailed test depends on the research question. When the research specifically is concerned with whether the true proportion exceeds a certain value (as in this case), a one-tailed test should be used. Using a two-tailed p-value calculation in such a context is incorrect because it splits the alpha level across both tails, diluting the evidence for a directional alternative hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the status quo or a specific value of the parameter, and the alternative hypothesis (Ha) represents what the researcher wants to provide evidence for. Here, the null hypothesis should state that the true proportion p is 0.75, while the alternative hypothesis should be that p is greater than 0.75. Using an estimate (such as 0.797) in the alternative hypothesis is a mistake; the hypotheses must be defined in terms of the true parameter value, not the sample estimate.

Standard Error Calculation

The standard error in proportion testing is calculated using the null hypothesis proportion and the sample size, not an estimated sample proportion or a misinterpreted effective sample size. The correct formula is ?[p0(1 ? p0)/n] where p0 is the proportion under the null hypothesis and n is the total sample size. Using an inappropriate value for n or p0 leads to errors in the computed z-score and subsequent conclusions.

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