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Confidence Intervals:
The Basics
Statistic ± (critical value)* (standard deviation of statistic)
Section 8.1
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
From Chapter 7 to Chapter 8
We are making a transition.
In chapter 7 we assumed that we knew the true value of a
parameter and then asked questions about the distribution of the
statistic used to estimate that parameter.
Now, in chapter 8 we no longer pretend to know the true
value of a parameter. We start with the more realistic
situation where we know only the value of a sample
statistic. Then we use that to estimate the parameter.
Objectives
• Point Estimator/ Point Estimate
• Idea Of Confidence Intervals and Levels
– Confidence Interval
– Confidence Level C
• Applet! Lets explore!
• Calculating Confidence Intervals
• Conditions of Constructing Confidence
Intervals
So…
• Statisticians are never 100% confident in their
results. So taking into account of any
variability and eliminate any possibility to be
proven wrong with one example that’s outside
of our results, we (Statisticians) use
confidence intervals. These intervals are used
to describe a specific range of low and high
numbers with a certain percent of certainty
depending on how large of a gap we are
leaving between numbers!
Can We Use 𝑥?
WHY?
We can use the value of the statistic 𝑥 because
the value 𝑥 is an unbiased estimator of the
population mean µ.
Point Estimator / Point estimate
• If we had to give a single number to estimate
the value of the statistic 𝑥 what would it be?
(such a value is known as a point estimate).
Example: “Jenny, if you had to guess what the sample mean
grade of your brother’s Algebra 1 class would be, what would
you guess?”
• Point estimator is a statistic that provides an
estimate of a population parameter. Point estimate
is the number.
Ideally, a point estimate is our “best guess” at the
value of an unknown parameter.
Intro To Confidence Intervals
• Example: Jenny, if you had to guess what the sample mean
of your brother’s Algebra 1 class would be, what would you
guess?
• She says:______
• I say, “now you don’t imagine that’s exactly the score, could
you give us a low score and a high score that you think the
class mean would fall between?”
Question of the day is:
How confident are you with that interval you
gave me? Do you want to be 95% confident?
Making connections…
• In Chapter 2 we learned about the Empirical rule:
– 68% of values lie within (+1) σ of the mean
– 95% of values lie within (+2) σ of the mean
– 99.7% of values lie within (+3) σ of the mean.
• In Chapter 4, we learned if we randomly select the sample, we should be
able to generalize our results to the population of interest.
• In Chapter 7 we learned that if we take multiple samples each may vary
by a certain amount, but if we take all the possible combinations of
samples, then, we can construct a sampling distribution.
Recall Our Bell Curve
Confidence Intervals and Levels
If I want to be 95% confident, I would need to
create an interval that is +2σ and -2σ
A confidence interval for a parameter has 2 parts:
Confidence Interval
Confidence level
“interpret the confidence interval…”
“interpret the confidence level…”
Confidence Interval
1) An interval calculated from the data, which has the
form for 95% CI:
Statistic ± (critical value)* (standard deviation of statistic)
“We are 95% confident that the interval from ____ to ____
captures the true [parameter in context]”
**don’t worry about calculating critical value for now**
Estimate ± margin of error
The margin of error tells how close the estimate tends to be to the
unknown parameter in repeating random sampling
Confidence Level
2) A confidence level C, which 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
“If we take many samples of the same size, (n), from
this population, about 95% of them will result in an
interval that captures the true [parameter value]”
Example: Do You Use Twitter?
In late 2009, the Pew Internet and American Life Project
asked a random sample of 2253 U.S. adults, “Do you
ever…use Twitter or another service to share updates
about yourself or to see updates about others?” Of the
sample, 19% said “Yes.” According to Pew, the resulting
95% confidence interval is (0.167, 0.213).2
PROBLEM: Interpret the confidence interval and the
confidence level.
Interval:
We are 95% confident that the interval from 0.167 to
0.213 captures the true population proportion of all US
adults who use Twitter or another service for updates
Example: Do You Use Twitter?
In late 2009, the Pew Internet and American Life Project asked a
random sample of 2253 U.S. adults, “Do you ever…use Twitter or
another service to share updates about yourself or to see
updates about others?” Of the sample, 19% said “Yes.”
According to Pew, the resulting 95% confidence interval is
(0.167, 0.213).2
PROBLEM: Interpret the confidence interval and the
confidence level.
Level:
If we take many samples of 2253 US adults, 95% of
confidence intervals will result in an interval that
captures the true population proportion of US adults
who use twitter or other services to share updates about
themselves or others.
CHECK YOUR UNDERSTANDING
How much does the fat content of Brand X hot dogs
vary? To find out, researchers measured the fat content
(in grams) of a random sample of 10 Brand X hot dogs.
A 95% confidence interval for the population standard
deviation σ is 2.84 to 7.55.
1. Interpret the confidence interval.
2. Interpret the confidence level.
3. True or False: The interval from 2.84 to 7.55 has a
95% chance of containing the actual population
standard deviation σ. Justify your answer.
Simulating Confidence Intervals
http://www.rossmanchance.com/applets/ConfSi
m.html
Calculating a Confidence Interval
statistic ± (critical value) · (standard deviation of statistic)
Lets just get some straight up Algebra number crunching
down for now, we will most definitely elaborate further with
more context!
Construct a 95% confidence interval.
𝒙 = 𝟐𝟒𝟎. 𝟕𝟗
𝝈𝒙 = 𝟓
Critical value = 1.96
statistic ± (critical value) · (standard deviation of statistic)
Conditions for Constructing A
Confidence Interval
Random:
The data should come from a well-designed random sample or
randomized experiment.
Normal:
Constructing confidence intervals should come from a sampling
distribution that is at least approximately normal.
– For Pop.Means: if pop is normal, sample is normal. If pop is
not normal the CLT of a sample greater than or equal to 30.
– For Pop.Proportions: normal conditions checked
Independent:
The procedure of calculating confidence intervals
assume that individual observations are independent.
Objectives
• Point Estimator/ Point Estimate
• Idea Of Confidence Intervals
– Confidence Interval
– Margin Of Error
– Confidence Level C
• Applet! Lets explore!
• Calculating Confidence Intervals
• Conditions of Constructing Confidence
Intervals
Homework
8.1 Homework Worksheet
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