One-sample Proportion Practice

Test Requirements

Recall that we can use the normal distribution to approximate the distribution of \(\hat{p}\) under certain conditions.

Question: What to we need to check before we assume normality for a hypothesis test?
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Question: What to we need to check before we assume normality for confidence intervals?
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Teenage Smoking Habits

A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. Previous surveys showed that 15% percent of all teenagers smoke. A Gallup survey interviewed a nationally representative sample of 785 teenagers aged 13 to 17. Seventy-one teenagers in the survey acknowledged having smoked at least once in the past week.

We want to explore if the study provide adequate evidence to conclude that the percentage of teenagers who smoke is now different than 15%.

Question: Are the requirements for both a confidence interval and a hypothesis test met and why?
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Whether or not the test requirements are met, perform the appropriate hypothesis test and create a confidence interval for the true proportion of teenagers who smoke.

Hypothesis Test

State the null and alternative hypotheses (replace the ??? with the correct information):

\[H_0: p=???\]
\[H_a: p???0.15\] Choose your confidence level:

\[\alpha = 0.\]

Question: What is the value of the test statistic?
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Question: What is the P-value?
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Question: Based on your selected \(\alpha\) and P-value, what is your conclusion?
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Confidence Interval

Create a 99% confidence interval for the true population proportion of teenagers who smoke?

Sample Size Calculations

During the design phase of the statistical process, we need to determine how much data will need to be collected to get “good” results. This is often a messy process and requires some assumptions. First, we need to know what “good enough” means. This can be determined by deciding how big we want our Margin of Error to be. For example, we may want to be within 3% of the truth.

Recall that the margin of error is equal to some \(z^*\) multiplied by standard error of the test statistic:

\[ME = z^* \sqrt{\frac{p(1-p)}{n}}\]

Solving for \(n\) after some algebra we get:

\[n=\left(\frac{z^*}{ME}\right)^2p(1-p)\]

This means that how much data we need actually depends on what the population proportion is and how small we would like the Marge in Error (ME) to be.

But how are we supposed to know \(p\) before we do the study?

Situation 1: Prior Estimate

We can use prior estimates from former studies or a subject matter expert’s estimate of \(p\) during the design phase of the study.

Situation 2: No Prior Estimate

If we have no idea what \(p\) is prior to the study, we can safely use \(p=0.5\) which makes \(p(1-p)\) as large as possible. This will give a conservative estimate of how much data we need.

You can also think of it is that if you don’t have a clue what the population proportion, \(p\), is ahead of time, treat it like a coin toss: 50/50.

Use this information to answer the questions below.

Perscription Use

A survey of doctors is planned to see what percentage prescribe a certain medication. Find the sample size required to achieve a 2% margin of error if the confidence level is 95%. Assume there are no prior estimates for p.

Question: How many doctors would we need to survey?
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Proportion of Left-handed Artists

Suppose you would like to study creativity of left-handed people. Your favorite generative AI told you that the estimated population of left-handed people in the United States is 11%. We would like to have a Margin of Error less \(\pm3\%\) (0.03) with 90% confidence.

Question: How many artists would you have to survey?
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