Download Confidence Intervals

TEKS
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(6.10) Probability and statistics. The student uses statistical
representations to analyze data.
The student is expected to:
(B) identify mean (using concrete objects and pictorial
models), median, mode, and range of a set of data;
(7.11) Probability and statistics. The student understands that the
way a set of data is displayed influences its interpretation.
The student is expected to:
(B) make inferences and convincing arguments based on an
analysis of given or collected data.
(7.15) Underlying processes and mathematical tools. The student
uses logical reasoning to make conjectures and verify conclusions.
The student is expected to:
(B) validate his/her conclusions using mathematical
properties and relationships.
(8.13) Probability and statistics. The student evaluates predictions
and conclusions based on statistical data.
The student is expected to:
(B) recognize misuses of graphical or numerical information
and evaluate predictions and conclusions based on data
analysis.
NCTM Standards
Grades 6-8 Expectations
• Select and use appropriate statistical methods to
analyze data
- Find, use, and interpret measures of center
and spread, including mean and interquartile
range
• Develop and evaluate inferences and predictions
that are based on data
- Use observations about differences between
two or more samples to make conjectures about
the populations from which the samples were
taken
Definitions to Know
• Population - the entire group of
individuals that we want information
about
• Sample - the part of the population
that we actually examine in order to
gather information
• Random Sampling - a selection that is
chosen randomly
• Sampling Distribution - the probability
distribution of a given statistic
More Definitions to Know
• Confidence Interval - a statistical range
with a specified probability that a
given parameter lies within the range
• Confidence Level - the level of
certainty to which an estimate can be
trusted
• Standard Deviation - a measure of how
spread out the data is
• p - a known or given “true” proportion
• p̂ - sample population proportion
Normal Distribution
A basic fact of normal distribution is
that 95% of all observations lie
within two standard deviations on
either side of the mean.
Normal Distribution
So, if p̂ lies within
two standard
deviations of the
true proportion in
95% of the
samples, we can
say that we are
95% confident
that the unknown
population
proportion lies
within a certain
interval.
What do Confidence
Statements Mean?
“We got these numbers by a method that gives
correct results 95% of the time.”
The confidence interval can either –
1. Contain the true population proportion or
2. Not contain the true population proportion
• We cannot know if our sample is one of the
95% for which the interval catches p or one of
the unlucky 5%.
Significance Test
• Used to assess whether an effect or
difference is present in the population
• Answers the question: “Is the
observed effect larger than can
reasonably be attributed to chance
alone?”
• Uses a correlation coefficient, r, to
show if there is a relationship between
the two variables and how strong it is
Steps for Calculating a 95%
Confidence Interval
1. Calculate the mean,
- Average data collected
2. Calculate the standard deviation, σ
- Subtract the mean from every number to
get the list of deviations
- Square the resulting list of numbers
- Add up all of the resulting squares to get
their total sum
- Find the mean of this sum, this is the
variance
- Find the square root of the variance
Steps for Calculating a 95%
Confidence Interval
3. Calculate the standard deviation of
the sampling distribution (standard
error)
=
4. Calculate the confidence interval
=
*1.96 comes from the Z-table and refers to the area of 2 standard
deviations from the mean. 1.96 is always used for calculating
the 95% confidence interval.
Demonstration Activity
Use the data on the next slide to calculate
an estimate for the true mean sales of
the ten highest selling box office
movies.
Then create a confidence interval to back
up your estimate.
Demonstration Activity
Rank
Movie
Sales (in
millions)
1
Titanic (1997)
600
2
The Dark Knight (2008)
530
3
Star Wars (1997)
460
4
Shrek 2 (2004)
440
5
E.T.: The Extra-Terrestrial (1982)
435
6
Star Wars: Episode I - The Phantom Menace (1999)
430
7
Pirates of the Caribbean: Dead Man's Chest (2006)
425
8
Spider-Man (2002)
400
9
Star Wars: Episode III - Revenge of the Sith (2005)
380
10
The Lord of the Rings: The Return of the King (2003)
380
Data retrieved from: The Internet Movie Database
http://www.imdb.com/boxoffice/alltimegross
Calculating True Mean of Sales
1. Calculate the mean,
448 millions of dollars
2. Calculate the standard deviation, σ
- Subtract the mean from every number to get the list of deviations
152, 82, 12, -8, -13, -18, -23, -48, -68, -68
- Square the resulting list of numbers
23104, 6724, 144, 64, 169, 324, 529, 2304, 4624, 4624
- Add up all of the resulting squares to get their total sum
42610
- Find the mean of this sum, this is the variance
4261
- Find the square root of the variance, yielding the standard deviation
65.28
Calculating Confidence Interval
3. Calculate the standard deviation of the
sampling distribution
=
= 20.64
4. Calculate the confidence interval
= 448 ± (1.96*20.64)
= (407.55, 488.45)
What is our confidence statement?
We are 95% confident that the true mean of sales for
the ten highest selling box office movies is contained
in the above confidence interval.
Possible Sources of Error
• As noted at the bottom of the website:
- Figures are not adjusted for inflation.
- Some movies may still be in general
release; all figures are estimated and
subject to change.