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Numerade Educator

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Problem 47 Hard Difficulty

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.

Answer

There is sufficient evidence to support the claim.

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Video Transcript

it's this question asked you, ah, to spot them steaks in this high pots his test, and the first mistake that should stick out is, um, we see that H O is given as p equals 0.75 which is right. That's what you're no Apophis. It should be. But your old kind of politics system should never have a value that is different. Some the null hypothesis that should not be 0.7 on seven. That doesn't make any sense, right? That's what about values in between these two. All right, so it's gonna be P is greater than 0.75 And so this changes are busy statistic which we're going to calculate by, um, doing the, um, one that we observed are observed value, which is 10.797 with 97 minus point 75 Divided by our standard deviation, which is also calculated wrong. I know that the standard deviation is P becomes one minus the over and and it's the square root of that whole value. So if you see how they did it, they used the 0.797 times 0.2 of three. But you actually want to use 30.75 point 25 and then used it for 45. Um, but is it for 45? No, you're an is 558 students. This is gonna be 5 58 This is gonna be point to 5.75 and this is gonna be point to five. So those are the mistakes, and those are going to change the values that you get. You also want to make sure if you're doing P is greater than 0.75 it's not.