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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.

Step by step solution

01

Identify the Correct Hypotheses

The first mistake is in the hypotheses. The null hypothesis \( H_0 \) should be \( p = 0.75 \), and the alternative hypothesis \( H_a \) should be \( p > 0.75 \), not \( \hat{p} > 0.797 \). This is because we want to test if more than three-quarters (75%) of students are engaging in bullying, so we compare to 0.75.

02

Correction of Proportion Calculation

The sample proportion \( \hat{p} \) should be calculated as the number of students exhibiting the behavior divided by the total number of students surveyed: \( \hat{p} = \frac{445}{558} \approx 0.797 \). This calculation is correct in the context but should be used with the correct null value comparison.

03

Setup the Correct Test Statistic

Use the test statistic formula for proportions: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \( p_0 = 0.75 \), \( \hat{p} = 0.797 \), and \( n = 558 \). This results in: \[ z = \frac{0.797 - 0.75}{\sqrt{\frac{0.75(0.25)}{558}}} \].

04

Calculate the Correct Test Statistic

Substitute the correct values into the formula: \( z = \frac{0.797 - 0.75}{\sqrt{\frac{0.75 \times 0.25}{558}}} \approx 3.34 \).

05

Find the p-value

Using a standard normal distribution table, the p-value corresponding to \( z = 3.34 \) for a one-tailed test is very small (close to 0), indicating strong evidence against the null hypothesis.

06

Conclusion Based on p-value

Since the p-value is much smaller than the typical significance level (such as 0.05), we reject the null hypothesis. There is strong evidence that more than three-quarters of the middle school students engage in bullying behavior.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether to support or reject a proposed explanation about a data set. It starts with two hypotheses: the null hypothesis

• The null hypothesis \( H_0 \) suggests that there is no effect or no difference, and it represents the status quo.
• The alternative hypothesis \( H_a \) suggests that there is an effect or a difference, challenging the status quo.

To determine which hypothesis is more likely to be true, we examine the available data and calculate a test statistic. A significant test statistic helps decide whether to reject the null hypothesis, thus potentially supporting the alternative hypothesis.
For example, in the study of bullying, Maurice needs to test if more than three-quarters of students are involved in such behavior. His null hypothesis is "the proportion of students involved in bullying is 75%" and the alternative hypothesis is "the proportion is greater than 75%." Correctly setting these hypotheses is crucial since they form the foundation of the test. Ensuring that they accurately represent the situation is the first step towards making valid conclusions.

Sample Proportion

The sample proportion is a crucial aspect of hypothesis testing as it offers an estimate of what is happening in the entire population based on the sample data. It is simply the ratio of the number of successes (or occurrences of interest) in the sample to the total number of observations.
In the bullying study, the sample proportion \( \hat{p} \) is calculated by dividing the number of students who reported aggressive behaviors (445) by the total number of students surveyed (558). Thus, \( \hat{p} = \frac{445}{558} \approx 0.797 \).
A precise calculation of the sample proportion is vital because it serves as the best estimate of the population proportion. Once determined, it helps to calculate the test statistic for the hypothesis test. It's important to note that, while the sample proportion provides valuable insight into the population, the sample size and randomness of the sample are also critical factors influencing reliability.

Bullying Statistics

Bullying statistics provide insights into the prevalence and nature of aggressive behaviors in school settings. Understanding these statistics supports institutions in developing effective intervention programs.
In Maurice's exercise, the statistics collected reflect how many students report engaging in behaviors tantamount to bullying, such as physical aggression and name-calling.
By calculating the proportion of students exhibiting these behaviors, Maurice aims to determine whether over 75% are involved. Statistics like these are crucial for grasping the scale of bullying and instituting appropriate policy measures. Policies could then be adjusted based on whether the true proportion indicates a significant issue, which is exactly what hypothesis testing seeks to uncover. Addressing bullying with accurate data helps foster a safer school environment.

Z-test

The Z-test for a proportion is used to determine if a sample proportion significantly differs from a hypothesized population proportion. The Z-test compares the observed data to what we would expect under the null hypothesis.
The formula for the Z-test statistic in this context is: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \( \hat{p} \) is the sample proportion, \( p_0 \) is the null hypothesis proportion, and \( n \) is the sample size.
In handling Maurice's data, substituting \( \hat{p} = 0.797 \), \( p_0 = 0.75 \), and \( n = 558 \) into the formula yields a Z-value. A Z-value indicates how many standard deviations a data point is from the mean. A high absolute Z-value means the sample data significantly deviates from the null hypothesis.
This Z-value is then used to find a p-value, which tells us how likely it is to observe such data under the null hypothesis. If the p-value is small, like in Maurice's test, it suggests the data strongly contradicts the null hypothesis. The Z-test, therefore, provides a clear statistical approach to determine whether the claim about bullying behavior is supported by the sample data.