Brushing teeth, wasting water? A recent study reported that fewer than half of young adults turn off the water while brushing their teeth. Is the same true for teenagers? To find out, a group of statistics students asked an SRS of $60$students at their school if they usually brush with the water off. In the sample, $27$students said "Yes." The dotplot shows the results of taking $200$SRSS of $60$students from a population in which the true proportion who brush with the water off is $0.50$.

a. Explain why the sample result ($27$of the $60$students said "Yes") does not give convincing evidence that fewer than half of the school's students brush their teeth with the water off.

b. Suppose instead that $18$of the $60$students in the class's sample had said "Yes." Explain why this result would give convincing evidence that fewer than $50\%$of the school's students brush their teeth with the water off.

We need to find out the reason for the sample result ($27$of the $60$students said "Yes") does not give convincing evidence that fewer than half of the school's students brush their teeth with the water off.

As we know that $27$out of $60$students who answered "yes"

So, $\frac{27}{60}=0.45$

And We can see that there are many dots above $0.45$in the dotplot, indicating that obtaining a proportion of $0.45$who say "Yes" is quite likely.

Because the sample result is likely to occur, this means that there is no persuasive evidence that fewer than half of the school's children brush their teeth with the water turned off.

We need to find out the reason for the result that would give convincing evidence that fewer than $50\%$of the school's students brush their teeth with the water off.

As we know that $18$out of $60$students who answered "yes"

So,$\frac{18}{60}=0.3$

And, We can see that there are no dots above $0.30$or to the left of $0.30$in the dotplot, indicating that obtaining a proportion of $0.30$who respond "Yes" is extremely unlikely.

Because the sample result is exceedingly improbable to occur, this means that there is solid evidence that fewer than half ($50\%$) of the school's kids clean their teeth with the water turned off.