Section 10.1: Confidence Intervals AP Statistics Section 10.1: Confidence Intervals

Objective: To be able to understand the basic principles of confidence intervals. Points: Statistical Inference refers to methods for drawing conclusions about the population based on facts about a sample. Different samples will yield different results and perhaps different conclusions. Inference will be based on the behavior of sampling distributions. For instructional purposes only, we will assume that we know 𝜎 and we don’t know 𝜇. This is unrealistic.

Confidence Intervals: First type of formal inference Recall the 68-95-99.7 Rule: Reverse viewpoint: Interpreting confidence intervals: “I am 95% confident that the true average context is between a and b. Interpreting “confident”: We are confident in the method being used. If I were to repeat this process 100 times approximately 95 of my constructed confidence intervals will contain 𝜇. Confidence never refers to probability or chance.

Once a sample is taken, either our interval contains 𝜇 or it does not Once a sample is taken, either our interval contains 𝜇 or it does not. If it does then there is a probability of 1 that 𝜇 is in the interval and if it does not then there is a probability of 0 that it contains 𝜇. The typical confidence level is at least 90%. Example 1: Find the 95% confidence interval for the average number of licks it takes to get to the center of a Tootsie Pop. Assuming σ=178.46 𝑙𝑖𝑐𝑘𝑠 How do you feel about this interval?

The basic idea behind the method:

General Diagram for a Confidence Interval for 𝜇: C = the confidence level. It represents the central probability under the normal curve. 𝛼 = 1 – C. This represents the remaining combined area in the tails of the normal curve. 𝑧 ∗ (or as well will use 𝑧 𝛼 2 ) represents the critical value that corresponds to an area of 𝛼 2 in the upper tail. Meaning that 𝑧 𝛼 2 is the z-score with 𝛼 2 to its right. Diagram:

Example 1: Find the critical value for a 70% C. I Example 1: Find the critical value for a 70% C.I. Example 2: Find the critical value for a 82% C.I. Common critical values: General form for any confidence interval: ESTIMATE ± MARGIN OF ERROR (m) Where m = critical value ∙ standard error C 𝛼 2 𝑧 𝛼 2 90% 95% 99%

Standard error is the standard deviation of the sampling distribution of a statistic. A level C Confidence Interval for 𝝁 (z-interval for 𝝁) Conditions: Data is an SRS. Normality: a. The population is stated to be normal. OR b. The sampling distribution of 𝑥 is normal. Independence: a. Observations are independent. OR b. Population ≥ 10n General form: 𝑥 ± 𝑧 𝛼 2 𝜎 𝑛

Example: Create a more accurate 95% confidence interval for the number of licks it takes to get to the center of a tootsie pop. How Confidence Intervals Behave: Optimal scenario: High confidence with a low margin of error. Margin of error controls the width of the confidence interval. 𝑚=𝑧 𝛼 2 𝜎 𝑛

Ways to minimize m: 1. 𝑧 𝛼 2 : 2. n: 3. 𝜎:

Choosing a Sample Size (n): To determine the sample size n that will yield a confidence interval for a population mean 𝜇 with a specified margin of error m, solve the following inequality for n: 𝑚≥𝑧 𝛼 2 𝜎 𝑛 Example: If we want to create a 95% confidence interval for the number of licks it takes to get to the center of a tootsie pop with a margin of error of 10 licks and a standard deviation of 178.46, how many lollipops are required?

Points of Caution with Confidence Intervals: Data must be an SRS. Different sampling techniques need different methods for analysis. Fancy formulas can’t fix poorly collected data. Outliers can distort the results. Are confidence intervals for means resistant? Explain. Shape of the population matters We know 𝜎. (unrealistic) Margin of error only covers random sampling errors. It deals with how much error can be expected by chance variation. Undercoverage and nonresponse can add additional error to m.