91Ó°ÊÓ

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 "No." The Fathom dotplot 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) Explain why the sample result 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 students in the class's sample had said "No." Explain why this result would give strong evidence that fewer than \(50 \%\) of the school's students brush their teeth with the water off.

Step by step solution

01

Understand the Hypotheses

To solve this problem, first establish the hypotheses. The null hypothesis (H0) is that the true proportion of students who turn off the water is 0.50. The alternative hypothesis (Ha) is that the true proportion is less than 0.50.

02

Calculate the Sample Proportion

The sample proportion (p̂) is calculated by dividing the number of students who do not turn off the water by the total number of students sampled. For 27 students out of 60: \[\hat{p} = \frac{27}{60} = 0.45.\] This gives a sample proportion of 0.45.

03

Discuss Sample Result for Part (a)

Compare the sample proportion with the comparison distribution. Since the true proportion is hypothesized to be 0.50 and the sample proportion is 0.45, this difference isn't substantial enough to reject the null hypothesis, when considering the variability of random samples. In various simulations of random samples from a population where the true proportion is 0.50, a result like this happens often enough that we cannot assert a different population proportion with confidence.

04

Calculate for Hypothetical Sample in Part (b)

Suppose only 18 students out of 60 said \"No\" to turning off the water:\[\hat{p} = \frac{18}{60} = 0.30.\]This sample proportion is 0.30.

05

Explain Result for Part (b)

A sample proportion of 0.30 is much further from the hypothesized true proportion of 0.50. In a typical sampling distribution derived from a population proportion of 0.50, such an extreme low value provides substantial evidence against the null hypothesis. Since such a result is much less likely to occur just by chance, it indicates strong evidence that fewer than 50% of students turn off the water.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion

A sample proportion is a way to estimate the true proportion of a population. In our exercise, we calculated the sample proportion as the number of students who say "No" to turning off water while brushing divided by the total students sampled. For example, when 27 out of 60 students said "No," we find:\[ \hat{p} = \frac{27}{60} = 0.45 \]This means 45% of our sample leaves the water running. Calculating the sample proportion helps to make inferences about the entire population from which the sample is drawn. By comparing the sample proportion to a hypothesized population proportion, we can make conclusions about the population's behavior.

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a statement of no effect or no difference in the context of an experimental setup. In this context, the null hypothesis is that the true proportion of the school's students who turn off the water is 0.50. We consider this to be a starting point.

• The null hypothesis assumes that the observed data result from just random variation.
• For this problem, \(H_0: p = 0.50\).
• We test this hypothesis by examining how much our sample proportion differs from 0.50.

The essence of the null hypothesis is to provide a benchmark by which we can compare our observed sample proportion.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_a\), proposes a different scenario than the null hypothesis. For this problem, it suggests that the true proportion is less than 0.50, meaning fewer than half the students turn off the water.

• It challenges the status quo represented by the null hypothesis.
• For this exercise, \(H_a: p < 0.50\).
• If our sample proportion is significantly lower in a statistical sense, we consider \(H_a\) as more plausible.

In hypothesis testing, the alternative hypothesis is where we look for evidence to support our claim against the null.

Sampling Distribution

A sampling distribution is crucial for understanding how sample statistics, like the sample proportion, behave across different samples. It helps provide a basis for comparison and is instrumental in hypothesis testing.

• A sampling distribution is the distribution of a statistic (\( \hat{p} \)) if we were to take many samples of the same size from the same population.
• It shows us what values our sample proportion might take just due to random sampling differences.
• In this exercise, a normal distribution underlies the sampling distribution when the number of samples tends to be large.

Understanding this concept helps us evaluate whether our observed sample proportion is a likely occurrence under the assumed population proportion, or if it provides evidence against it.