Let's return to our confidence interval that was given before:
We are estimating a 95% confidence interval of what proportion of high school math students pass their class. We were given a sample of 85 students where ~75% of them passed. We calculated a confidence interval for the true population proportion based off of our sample. The interval is given in the calculator output below:
We are given the 95% confidence interval (0.66125, 0.84463) as an estimate of the population proportion of high school students who are passing their math class.
In terms of what this means, it means we are 95% confident that the true population proportion of high school students who pass their math class is between 0.66125 and 0.84463. Notice that both of our endpoints are decimals less than 1. This is because we are estimating a proportion, which is always between 0 and 1. Anytime we are calculating any type of proportion, our answer should always be between 0 and 1.
image courtesy of: pixabay.com
When interpreting a confidence interval for a population proportion, there are three things necessary to receive full credit: confidence level, context, and reference to true population proportion.
Our confidence level is generally given in the problem. This is the 95%, 90%, 98%, etc. This impacts the z* for our confidence interval and is necessary in including in our interpretation of the interval.
As with anything in AP Stats, context is essential to receive full credit. Anytime we write out an answer, we need to include it in context of the problem being asked. It is no different when interpreting a confidence interval. We need to ask ourselves, "What is this interval estimating?" and include that in our response.
We also need to be sure that our answer implies that we are estimating a population proportion, not just a sample proportion. After all, there's no reason to estimate something for our sample because we have the EXACT sample proportion as it was given to us. We are using that sample to estimate the bigger picture with our population. 😊
When we are given a population proportion that maybe we don't necessarily believe, we can use a confidence interval based off of a random sample to test that claim. The main way we are going to check the statistical claim is by seeing if the claimed population proportion is within our confidence interval. If it is in our confidence interval, then it is possible that the claim is true. If the claimed value is NOT in our interval, we may need to investigate further to see if the claim made by an article/study is in fact false.
In our example above dealing with students passing their math class, let's say that we recently read an article that said only 55% of all US students are passing their math class. Therefore, we took a random sample of 85 US math students and we were given the interval above: (0.66125, 0.84463).
Since 0.55 is NOT in our interval, we have reason to doubt the article that we read. We should definitely investigate it further. 🕵️🕵️
When we are given a claim that we are checking, our expected successes and failures change for our Large Counts Condition that we checked in Unit 6.2. Now that we are given a supposed proportion to be true for the population, we use that to calculate our expected successes and failures. So our large counts condition would change to 0.55(85)≥10 & 0.45(85)≥10, which still holds for this particular problem.
In other words, when we are given an actual p to check this condition, use it. When we aren't given a p-value, use the next best thing by using your p-hat.
🎥Watch: AP Stats - Inference: Confidence Intervals for Proportions
✍️ Free Response Questions (FRQs)
👆 Unit 1: Exploring One-Variable Data
1.4Representing a Categorical Variable with Graphs
1.5Representing a Quantitative Variable with Graphs
1.6Describing the Distribution of a Quantitative Variable
1.7Summary Statistics for a Quantitative Variable
1.8Graphical Representations of Summary Statistics
1.9Comparing Distributions of a Quantitative Variable
✌️ Unit 2: Exploring Two-Variable Data
2.0 Unit 2 Overview: Exploring Two-Variable Data
2.1Introducing Statistics: Are Variables Related?
2.2Representing Two Categorical Variables
2.3Statistics for Two Categorical Variables
2.4Representing the Relationship Between Two Quantitative Variables
2.8Least Squares Regression
🔎 Unit 3: Collecting Data
3.5Introduction to Experimental Design
🎲 Unit 4: Probability, Random Variables, and Probability Distributions
4.1Introducing Statistics: Random and Non-Random Patterns?
4.7Introduction to Random Variables and Probability Distributions
4.8Mean and Standard Deviation of Random Variables
4.9Combining Random Variables
4.11Parameters for a Binomial Distribution
📊 Unit 5: Sampling Distributions
5.0Unit 5 Overview: Sampling Distributions
5.1Introducing Statistics: Why Is My Sample Not Like Yours?
5.4Biased and Unbiased Point Estimates
5.6Sampling Distributions for Differences in Sample Proportions
⚖️ Unit 6: Inference for Categorical Data: Proportions
6.0Unit 6 Overview: Inference for Categorical Data: Proportions
6.1Introducing Statistics: Why Be Normal?
6.2Constructing a Confidence Interval for a Population Proportion
6.3Justifying a Claim Based on a Confidence Interval for a Population Proportion
6.4Setting Up a Test for a Population Proportion
6.6Concluding a Test for a Population Proportion
6.7Potential Errors When Performing Tests
6.8Confidence Intervals for the Difference of Two Proportions
6.9Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions
6.10Setting Up a Test for the Difference of Two Population Proportions
😼 Unit 7: Inference for Qualitative Data: Means
7.1Introducing Statistics: Should I Worry About Error?
7.2Constructing a Confidence Interval for a Population Mean
7.3Justifying a Claim About a Population Mean Based on a Confidence Interval
7.4Setting Up a Test for a Population Mean
7.5Carrying Out a Test for a Population Mean
7.6Confidence Intervals for the Difference of Two Means
7.7Justifying a Claim About the Difference of Two Means Based on a Confidence Interval
7.8Setting Up a Test for the Difference of Two Population Means
7.9Carrying Out a Test for the Difference of Two Population Means
✳️ Unit 8: Inference for Categorical Data: Chi-Square
📈 Unit 9: Inference for Quantitative Data: Slopes
🧐 Multiple Choice Questions (MCQs)
Is AP Statistics Hard? Is AP Statistics Worth Taking?
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