1. CHAPTER 10 Estimating with Confidence
10.1
Confidence Intervals:
The Basics

2. Confidence Intervals: The Basics
DETERMINE the point estimate and margin of error from a confidence interval.
INTERPRET a confidence interval in context.
INTERPRET a confidence level in context.
DESCRIBE how the sample size and confidence level affect the length of a confidence interval.
EXPLAIN how practical issues like nonresponse, undercoverage, and response bias can affect the interpretation of a confidence interval.

3. Activity: The Mystery Mean
Suppose your teacher has selected a “Mystery Mean” value µ and stored it as “M” in their calculator. Your task is to work together with 3 or 4 students to estimate this value. The following command was executed on their calculator: mean(randNorm(M,20,16))
The result was This tells us the calculator chose an SRS of 16 observations from a Normal population with mean M and standard deviation 20. The resulting sample mean of those 16 values was
Your group must determine an interval of reasonable values for the population mean µ. Use the result above and what you learned about sampling distributions in the previous chapter.
Share your team’s results with the class.

4. Confidence Intervals: The Basics
A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate.
Will our point estimate be the same as the population parameter?

5. The Idea of a Confidence Interval
The value population mean µ is probably not exactly ? However, since the sample mean is , we could guess that µ is “somewhere” around How close to is µ likely to be? To answer this question, we must ask another:

6. The Idea of a Confidence Interval
We are 95% confident that the true mean, M, is between to
**This is a 95% Confidence Interval**

7. The Idea of a Confidence Interval
estimate ± margin of error
A C% confidence interval gives an interval of plausible values for a parameter. The interval is calculated from the data and has the form
point estimate ± margin of error
The confidence level C gives the overall success rate of the method for calculating the confidence interval. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value.

8. Interpreting Confidence Intervals
To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval from _____ to _____ captures the actual value of the [population parameter in context].”

9. Interpreting Confidence Intervals
A recent Gallup Poll asked a sample if healthcare should be the government's responsibility.
Based on the poll, Gallup reported that the 95% confidence interval for the population proportion who answered "no" was (0.47, 0.55)
(a) Interpret the Confidence Interval
(b) What is the point estimate used to create the interval? What is the margin of error?
(c) Based on this poll, a political reporter claims that fewer than half of Americans believe that healthcare should not be the governments responsibility. Use the confidence interval to evaluate this claim.
When the same poll was given in 2014, the 95% confidence interval for the population who answered "no" was (0.41, 0.49).
(d) Based on this poll, could we claim that fewer than half of Americans in 2014 believed that healthcare should not be the governments responsibility?

10. Interpreting Confidence Levels
The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times.
The confidence level does not tell us the chance that a particular confidence interval captures the population parameter.

11. Interpreting Confidence Levels
To say that we are 95% confident is shorthand for “If we take many samples of the same size from this population, about 95% of them will result in an interval that captures the actual parameter value.”

12. EX: Keep on Walkin’
A large company is concerned that many of its employees are in poor physical condition, which can result in decreased productivity. To determine how many steps each employee takes per day, on average, the company provides a pedometer to 50 randomly selected employees to use for one 24-hour period. After collecting the data, the company statistician reports a 95% confidence interval of 4547 steps to 8473 steps.
Interpret the confidence level.
Interpret the confidence interval.

13. How Confidence Intervals Behave
1) Larger confidence =
Lower confidence =
2) Larger sample size =
Smaller sample size =

14. Discuss with a partner :
I am designing a study, and I want to ensure that I have a small margin of error.
(a) What are two ways that I can ensure this?
(b) What are the pros and cons of each option?

15. Constructing Confidence Intervals
When we calculated a 95% confidence interval for the mystery mean µ, we started with
estimate ± margin of error
This leads to a more general formula for confidence intervals:
statistic ± (critical value) • (standard deviation of statistic)

16. Using Confidence Intervals Wisely
Here are two important cautions to keep in mind when constructing and interpreting confidence intervals.
Our method of calculation assumes that the data come from an SRS of size n from the population of interest.
The margin of error in a confidence interval covers only chance variation due to random sampling or random assignment.

17. Confidence Intervals: The Basics
DETERMINE the point estimate and margin of error from a confidence interval.
INTERPRET a confidence interval in context.
INTERPRET a confidence level in context.
DESCRIBE how the sample size and confidence level affect the length of a confidence interval.
EXPLAIN how practical issues like nonresponse, undercoverage, and response bias can affect the interpretation of a confidence interval.