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Unit 6

6.2 Constructing a Confidence Interval for a Population Proportion

3 min readjune 5, 2020

josh

Josh Argo


AP Statistics 📊

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What is a Confidence Interval?

A confidence interval is an interval of numbers used to estimate a population parameter. In the case of categorical data, we are aiming at estimating a population proportion. This interval is based on our sample proportion, our sample size and the sampling distribution of that given sample size. Our confidence interval is reliant on a confidence level, which impacts how confident we are that our interval contains the true population proportion. The standard confidence level is usually 95%.
As the confidence level increases, the width of our interval also increases.

Checking Conditions

✅ Random Sample

This reduces any bias that may be caused from taking a bad sample. When answering inference questions, it is always essential to make note that our sample was random, either by highlighting text on the exam, or by quoting the problem where it details its randomness. Without a random sample, our findings cannot be generalized to a population. This means that our scope of inference is inaccurate. There are no calculations that can fix an un-random, or biased, sample.

Independence

This ensures that each subject in our sample was not influenced by the previous subjects chosen. While we are sampling without replacement, if our sample size is not super close to our population size, we can conclude that the effect it has on our sampling is negligible. We can check this condition by questioning if it is reasonable to believe that the population in question is at least 10 times as large as our sample.A good way to state this when performing inference is to say, "It is reasonable to believe that our population (in context) is at least 10n".
For example, if we have a random sample of 85 teenagers math grades and we are creating a confidence interval for what the proportion of ALL teenagers passing their math class, we could state, "It is reasonable to believe that there are at least 850 teenagers currently enrolled in a math class."

✅Normal

This check verifies that we are able to use a normal curve to calculate our probabilities using either empirical rule or z scores. We can verify that a sampling distribution is normal using the Large Counts Condition, which states that we have at least 10 expected successes and 10 expected failures.
In the example listed above, let's say that we were given the proportion that 70% of all teenagers pass their math class. That means that with a sample of 85, 0.75(85)=63.75, which is greater than 10. We also have to check the complement by calculating 0.25(85)=21.25, which is also greater than 10.
Since both np and n(1-p) are greater than or equal to 10, we can conclude that the sampling distribution of our proportion will be approximately normal.

Calculating Interval

Calculating a confidence interval is based on two things: our point estimate and our margin of error.

Point Estimate

Our point estimate for a confidence interval used to estimate a population proportion is our sample proportion, or our p-hat. This is the middle point of our confidence interval.

Margin of Error

Our margin of error is the "buffer zone" of our confidence interval. This is what we add and subtract to our sample proportion to allow some room for error in our interval. It is based on two things: critical value (z-score) and standard deviation. Both of these things are heavily impacted by the sample size. The larger the sample size, the smaller the margin of error.

Formula

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-17%20at%2011.38-5ADeErrkHbxx.png?alt=media&token=f474098b-dbcf-4c7a-93b9-54d8be866371

Using Calculator

A much easier, more efficient way of calculating a confidence interval is to use a graphing calculator or other form of technology. When using a graphing calculator such as a Texas Instruments TI-84, you would select 1-Prop Z Interval from the Stats menu, enter your number of successes (x), sample size (n), and confidence level. Lastly, calculate and you will get the confidence interval you requested!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-17%20at%2011.41-Ur96U3AxjdcP.png?alt=media&token=43594e56-ed47-45e3-9253-3a5db1adec66

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