Transcript p.p chapter 8.1

Confidence Intervals:
The Basics
Section 8.1
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
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
So…
• Statisticians are never 100% confident in their
results. To 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!
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.
Can We Use 𝑥?
• We can use the value of the statistic 𝑥
because the value 𝑥 is an unbiased
estimator of the population mean µ.
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, would
you say somewhere around ____? Between what numbers,
low and high?”
• Question of the day is: How confident are
you with that interval you gave me? Do you
want to be 95% confident?
Recall Our Bell Curve
Confidence Interval
• 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 two parts:
1) An interval calculated from the data, which has the
form
Estimate ± margin of error
The margin of error tells how close the estimate tends to be to the
unknown parameter in repeating random sampling
Margin or error can be elaborated on for 95% CI:
Statistic ± (critical value)* (standard deviation of statistic)
𝒙 ± 𝟐 ∗ 𝝈𝒙
**don’t worry about calculating critical value for now**
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
Example: If many samples are taken and 95% confidence
intervals are constructed based on these samples, then about
95% of the intervals will capture the true parameter being
estimated.
“95% of all possible samples of a given size from this population
will result in an interval that captures the unknown parameter.”
Simulating Confidence Intervals
• http://www.rossmanchance.com/applets/C
onfSim.html
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.
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.
Calculating a Confidence
Interval
• Calculating a Confidence Interval
• The confidence interval for estimating a
population parameter has the form:
statistic ± (critical value) · (standard deviation of statistic)
𝒙 ± 𝟐 ∗ 𝝈𝒙
Conditions for Constructing A
Confidence Interval
• Random: The data should come from a welldesigned random sample or randomized experiment.
• Normal: Constructing confidence intervals should
come from a sampling dist. That is at least
approximately normal.
– For 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 populations: 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
Worksheet