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.
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.
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."
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 a confidence interval is based on two things: our point estimate and our margin of error.
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.
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.
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!
🎥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)
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