Question

Recycling Do most teens recycle? To find out, an AP Statistics class asked an SRS of 100 students at their school whether they regularly recycle. How many students in the sample would need to say “Yes” to provide convincing evidence that more than half of the students at the school recycle? The Fathom dot plot below shows the results of taking 200 SRSs of 100 students from a population in which the true proportion who recycle is 0.50.

(a) Suppose 55 students in the class’s sample say “Yes.” Explain why this result does not give convincing evidence that more than half of the school’s students recycle.

(b) Suppose 63 students in the class’s sample say “Yes.” Explain why this result gives strong evidence that a majority of the school’s students recycle.

Short answer

Part (a) There isn't enough information to back up the assertion that more than half of the students at the school recycle.

Part (b) There is enough information to support the allegation that more than half of the students at the school recycle.

Step-by-step solution

Part (a) Step 1. Given Information 

$n=100$ SRS size The number of pupils who claimed they recycle is $55$

We must decide whether there is persuasive evidence that more than half of the pupils at the school would indicate they recycle regularly using the Fathom dot plot, which displays the results of taking $200$ SRSs of $100$ students from a population in which the genuine proportion who recycle is $0.50$

Part (a) Step 2. Concept Used 

A simulation is a model of random behaviour that is usually carried out with random numbers.

Part (a) Step 3. Explanation     

It is insufficient proof that more than half of the students at the school recycle because: "Yes" was replied by $55$ out of $100$ pupils. $\frac{55}{100}=0.55$

We can see that the proportion of $0.55$has a lot of dots above it in the dot plot. As a result, a proportion of $0.55$ is quite likely to be obtained when the genuine proportion is $0.5$, and there is insufficient evidence to support the assertion that more than half of the school's students recycle.

Part (b) Step 1. Explanation  

Because $63$ out of $100$ students answered "Yes," this statistic strongly suggests that the majority of the school's students recycle $\frac{63}{100}=0.63$

We can see that the proportion of $0.63$ has only one dot above it and no dots to the right of it in the supplied dot plot. Thus, while the genuine proportion is $0.5$, a proportion of $0.63$ is quite likely to be obtained, and there is adequate evidence to support the assertion that more than half of the school's students recycle. As a result, the outcome.