1. 8.1

2. A Change in Direction
So far (last chapter) we knew (or assumed) the true population value, and then asked questions about the distribution of the statistic used to estimate it
Usually we don’t know the true parameter value
So now we’re going to stop assuming that we know the true value. Instead, we will start with the more realistic situation that we know only the value for the statistic, and we will use it to estimate the parameter (population) value

3. Estimating a Population Value
A Point Estimator is a statistic that provides an estimate of a population parameter
Ideally no bias, low variability
The value of a statistic that we use to estimate the parameter is called the point estimate

6. The Idea of a Confidence Interval
Hypothetical: we have a normally distributed sampling distribution with a standard deviation of 5 and a mean of 241
Sampling distribution, not population distribution
For now, let’s not worry about how we got that
If we use the rule, then we know that 95% of all samples must have means within 2 standard deviations of 241
So within 10 of 241
So 95% of all samples will have means between 231 and 251

7. Confidence Level vs Confidence Interval
Confidence Level: How confident we want to be
In the example on the previous slide, 95% of samples were between 231 and 251.
If we took another random sample, 95% chance that the mean would be in that range
So we are 95% confident that the “true” value is somewhere in that range
So our confidence level was 95%
If we were using an 80% confidence level instead of 95%, would the range be wider or narrower?

8. Confidence Level vs Confidence Interval
The confidence interval is the actual range itself
So the confidence level is how confident we want to be
The interval is the actual range of values

9. 95% confidence

10. Common Mistakes
CORRECT: It is correct to say that we are “95% confident” that the true value is in the interval
CORRECT: it is correct to say that 95% of samples will include the true value
INCORRECT: it is incorrect to say that there is a 95% chance that the true value is in the interval

11. An Example
Researchers measure the fat content (in grams) in a certain brand of hot dogs. A 95% confidence interval for the population standard deviation is 2.84 to 7.55
Interpret the confidence interval
True or false: the interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard deviation

12. Two ways to write a confidence interval
One way is to express it as a range:
231 to 251
231—251
231, 251
The other is as a margin of error
241 ± 10

13. Calculating a Confidence Interval
Statistic ± margin of error
Statistic ± (critical value)(st. dev of statistic)
The critical value is determined by whatever confidence level we want
We will talk more about critical values next class
The most common confidence level is 95%

14. Margin of error Statistic ± (margin of error)
Statistic ± (critical value)(St. dev of statistic)
So if we want a smaller margin of error (which we usually do, there are two ways to do that):
Decrease the standard deviation
Larger sample
Decrease the critical value
Lower confidence level

15. Requirements for calculating a confidence interval
RANDOM: The data must come from a random sample
Or, if it is an experiment, one that uses random assignment
Normal: The method that we are going to use to calculate a confidence interval depends on the sampling distribution being approximately normal
REMINDER: this does not necessarily mean that the population is normal
For proportions: 𝑛𝑝≥10, 𝑛(1−𝑝)≥10
For means:
If populations is normal
OR if sample size is large (at least 30)
Independent: Observations are independent
Sample is ≤ 10% of the population

16. Homework:
Page 481: 1-4, 9-13, 15, 16, 18, 21-24