Question

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. How many

students in the sample would need to say “No” to provide convincing evidence that fewer than half of the students at the school brush with the water off? The Fathom dot plot below 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) Suppose 27 students in the class’s sample say “No.” Explain why this result does not give convincing evidence that fewer than half of the school’s students brush their teeth with the water off.

(b) Suppose 18 students in the class’s sample say “No.” Explain why this result gives strong evidence that fewer than 50% of the school’s students brush

their teeth with the water off.

Short answer

Part (a) It is not sufficient evidence to that more than half of the school’s student brush their teeth with the water off .

Part (b) It is sufficient evidence to that more than half of the school’s student brush their teeth with the water off .

Step-by-step solution

Part (a) Step 1. Given Information    

The dot plot depicts the result of taking $200$ SRSs from $60$ kids from a population with a real proportion of $0.50$ brushing with the water off.

Part (a) Step 2. Concept Used 

We can't foresee the outcomes of a chance process, yet they have a regular distribution over a large number of repetitions. According to the law of large numbers, the fraction of times a specific event occurs in numerous repetitions approaches a single number. The likelihood of a chance outcome is its long-run relative frequency. A probability is a number between $0$ (never happens) and $1$ (happens frequently) (always occurs).

Part (a) Step 3. Explanation     

Because $27$ out of $60$ kids answered "No," it does not provide persuasive proof that more than half of the school's students recycle $27/60=0.45$ We can see that the proportion of $0.45$ has a lot of dots above it in the dot plot. Thus, a proportion of $0.45$ is quite likely to be obtained when the true proportion is $0.5$, and there is insufficient evidence to support the assertion that more than half of the school's students wash their teeth with the water turned off.

Part (b) Step 1. Explanation  

Because $18$ out of $60$ kids answered "Yes," this statistic strongly suggests that the majority of the school's students recycle $18/60=0.3$We can see that the proportion of $0.5$ has no dot above it and no dots to the left of it in the supplied dot plot. Thus, when the genuine proportion is $55$, it is extremely likely to achieve a proportion of $0.3$, and there is adequate evidence to support the assertion that more than half of the school's students brush their teeth with the water turned off.