Download 8.1 Confidence Intervals: The Basics

8.1 Confidence Intervals: The Basics
Objectives
SWBAT:
• 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.
How is this chapter different than Chapter 7?
Chapter 8 is like the time that 7 strangers were picked together to live under one
roof, and have their lives taped, to find out what happens when people stop being
polite, and start getting real…Real World, AP Statistics.
How confident would you be in saying that it will be exactly 35 degrees tomorrow?
Would you be more or less confident if you came up with a range of values?
• For example, how about if you said it would be between 30 and 40 degrees?
• What about if you said it would be between 20 and 50 degrees? How confident would
you be in this prediction?
• What would the problem be with coming up with too large of a range?
• Obviously we would be pretty confident in this interval. However, it doesn’t really tell
us anything about the weather.
• We don’t know if it will be hot or cold (only Katy Perry knows that).
What is a point estimate? Why is it called a point estimate?
If you had to give one number to estimate an unknown population
parameter, what would it be? If you were estimating a population
mean µ,you would probably use x. If you were estimating a
population proportion p, you might use pˆ . In both cases, you would be
providing a point estimate of the parameter of interest.
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.
• The point estimate is our single best guess for the value of a population parameter (think
about one single value on the number line). However, it almost certainly is not correct?
• Therefore, what’s our solution?
• Use an interval of values!
Explain the logic of confidence intervals.
Let’s say that from a population, we take a sample of size 16 that has a mean of
240.79 and a standard deviation of 5.
To estimate  , we can use x  240.79 as a point
estimate. We don’ t expect  to be exactly equal to x so we need to
say how accurate we think our estimate is.
· The 68 - 95 - 99.7 Rule tells us that in 95%
of all samples of size 16, x will be within 10
(two standard deviations) of m.
· If x is within 10 points of m, then m is
within 10 points of x .
Therefore, the interval from x -10 to x + 10 will "capture" m in about
95% of all samples of size 16.
If we estimate that µ lies somewhere in the interval 230.79 to 250.79, we’d
be calculating an interval using a method that captures the true µ in about
95% of all possible samples of this size.
What is a confidence interval?
The big idea : The sampling distribution of x tells us how close to m the
sample mean x is likely to be. All confidence intervals we construct will
have a form similar to this :
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 difference between the point estimate and the true
parameter value will be less than the margin of error in C% of
all samples.
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.
What is the margin of error? Why do we include the margin of error?
• Margin of error is how far we expect the sample statistic to vary from the
population parameter, AT MOST.
• We include margin of error to account for sampling variability – the fact that
different samples will give us different estimates.
• We do not include the margin of error to account for bias!
• Additionally, include a margin of error greatly increases our changes of being
correct. Think about the Garfield comic, but to a smaller degree.
How do you interpret a confidence interval?
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].”
So if constructing a 95% confidence interval, your interpretation would be:
“We are 95% confident that the interval from
to
contains the true
.”
We are 95% confident that the interval from 59% to 67% contains the true
proportion of American adults who support the death penalty.
How do you interpret a confidence level? In other words, what does it mean to
be 95% confident?
• Confidence Intervals Applet
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.”
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.
Instead, the confidence interval gives us a set of plausible values for the parameter.
We interpret confidence levels and confidence intervals in much the same way whether we are
estimating a population mean, proportion, or some other parameter.
Alternate Example: 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.
a) Interpret the confidence level.
If we were to take many, many samples and calculate many, many intervals, about 95% of the
intervals will capture the true mean number of steps taken per day for employees at this company.
b) Interpret the confidence interval.
We are 95% confident that the interval from 4547 to 8473 captures the true mean number of steps
taken per day for employees at this company.
c) What is the point estimate that was used to create the interval? The margin of error?
The point estimate is the midpoint of the interval: (4547+8473)/2= 6510. The margin of error is the
distance from the point estimate to each endpoint of the interval: 8473 – 6510 = 1963.
d) Recent guidelines suggest that people aim for 10,000 steps per day. Is there convincing evidence
that the employees of this company are not meeting the guideline, on average? Explain.
Because all the values in the interval are less than 10,000, there is convincing evidence that the
employees of this company are not taking 10,000 steps per day, on average. That is, there is
convincing evidence that the employees are not meeting the guideline.
What is the formula for calculating a confidence interval? Is this formula
included on the formula sheet?
Calculating a Confidence Interval
The confidence interval for estimating a population parameter has
the form
statistic ± (critical value) • (standard deviation of statistic)
where the statistic we use is the point estimator for the parameter.
Formula sheet! Boom!
Prescription for happiness:
How can we reduce the margin of error in a confidence interval? Why do we
want a small margin of error? Are there any drawbacks to these actions?
• One way to reduce the margin of error is to reduce the confidence level. The
drawback to this is you are less likely to get a correct estimate.
• Another way to reduce the margin of error is to increase the sample size. The
drawback to this is that when we are physically collecting the sample, a larger
sample size will be more costly.
• The benefit of a small margin of error is that we are more precise.
What are two important things to remember 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.
In a 2009 survey, researchers asked random samples of US teens and adults if they use social
networking sites. Overall, 73% of the teens said yes and 47% of the adults said yes. A 90%
confidence interval for the true difference in the proportion of teens and adults who would
say yes is 0.229 to 0.291.
a) Interpret the confidence level.
If we were to take many, many samples and calculate many, many intervals, about 90% of
the intervals will capture the true difference in the proportion of teens and adults who use
social networking sites.
b) Interpret the confidence interval.
We are 90% confident that the interval from 0.229 to 0.291 captures the true difference in
the proportion of teens and adults who use social networking sites.
c) Based on the interval, is there convincing evidence that the proportion of teens who
would say yes is higher than the proportion of adults who would say yes? Explain.
Yes because 0 is not contained in the interval. Every value in the interval has the proportion
of teens saying yes higher than the proportion of adults saying yes.
d) How would the interval be affected if we used a 99% confidence level instead of a 90%
confidence level?
The interval would be larger in order for us to be more confident.