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. In the sample, $55$students said that they recycle. Is this convincing evidence that more than half of the students at the school would say they regularly recycle? The dotplot shows the results of taking $200$SRSS of $100$students from a population in which the true proportion who recycle is $0.50$.

a. Explain why the sample result ($55$out of $100$said "Yes") does not give convincing evidence that more than half of the school's students recycle.

b. Suppose instead that $63$students in the class's sample had said "Yes." Explain why this result would give convincing evidence that a majority of the school's students recycle.

Short answer

(a) Many points are over the sample fraction of $0.55$of the dotplot.

(b) In the dotplot, there is just one dot above $0.63$and none to the right of $0.63$.

Step-by-step solution

Part (a) Step 1: Given information

We need to find out the reason for the sample result ($55$out of $100$said "Yes") does not give convincing evidence that more than half of the school's students recycle.

Part (a) Step 2: Explanation

As we know that $55$out of $100$students who answered "yes"

So, $\frac{55}{100}=0.55$

And, We can see that the proportion of $0.55$has a lot of dots above it in the dotplot.

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: Given information

We need to find out the reason for the result would give convincing evidence that a majority of the school's students recycle.

Part (b) Step 2: Explanation

As we know that $63$out of $100$students who answered "yes"

So,$\frac{63}{100}=0.63$

And 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 dotplot.

Thus, when the genuine proportion is $0.5$and there is adequate data to support the assertion that more than half of the school's pupils recycle, a proportion of $0.63$(or a more extreme proportion) is highly unlikely.