Topics for Today
t-Distribution
Review for Midterm
Stat203
Fall 2011 – Week 7 Lecture 1
Page 1 of 23
t-distribution
Bell-shaped but with thicker tails than a normal
distribution.
Used when we no longer ‘know’ .
Thickness of the tails (and therefore it’s
similarity to a Normal distribution) is controlled
by something called the
‘__________________’
In any particular situation, the degrees of
freedom are ___
… as the degrees of freedom get bigger, the tdistribution gets closer and closer to a ______
distribution.
… so … the larger our sample size (n), the
less _______ we’ll pay for using s instead of .
Stat203
Fall 2011 – Week 7 Lecture 1
Page 2 of 23
Confidence Interval for µ
when  is unknown
X  t n1 s
n
The margin of error for a 95% (α =0.05) confidence
interval for the population mean μ with a sample size of
______ is given by:
s
t
n 1
n
 t 24 s
25
 2.064 s 5  0.413s
For a 95% (α =0.05) confidence interval with a sample
size of _______ :
tn 1 s
n
 t 99 s
100
 1.984 s10  0.198s
For a __% (α =0.10) confidence interval with a sample
size of n = 100:
tn 1 s
n
 t 99 s
Stat203
Fall 2011 – Week 7 Lecture 1
100
 1.660 s10  0.166s
Page 3 of 23
Where did those tn-1 values come from?
Another table in the back of the textbook ..
Table C, Page 519.
Where  is part of the confidence level; as in
for a 95% confidence level,  = 5%
100% -  = confidence level
Stat203
Fall 2011 – Week 7 Lecture 1
Page 4 of 23
Stat203
Fall 2011 – Week 7 Lecture 1
Page 5 of 23
Example: Pg 204 #8
Standardized achievement test; observe 16
individuals. Construct a 95% CI.
mean = __
standard deviation = ____
standard deviation of mean = standard error of
mean = s/n ________________
 = __
____________________________(from Table C)
95% CI:
X  t n 1sx  13  (2.13)(1.73)
 13  3.68
 [9.32,16.68]
Stat203
Fall 2011 – Week 7 Lecture 1
Page 6 of 23
Another Example with  unknown
Let’s do a CI using SPSS, and one by hand to
compare.
Using the GSS data set, let’s calculate a 95%
confidence interval for the
__________________ in the United States.
Using the ‘Explore’ command we get the
following:
Stat203
Fall 2011 – Week 7 Lecture 1
Page 7 of 23
So, SPSS gives us the 95% CI right away!
By hand, we can use the other numbers in this
table to calculate our CI.
Stat203
Fall 2011 – Week 7 Lecture 1
Page 8 of 23
Interpreting Confidence Intervals
Think back to the NAEP example. The
confidence interval for the mean quantitative
score was:
The CI was x 4.2 ________________
Is ___ a reasonable guess at the true
population mean quantitative score?
Why or why not?
Is ___ a plausible value for the mean?
The CI gives us a range of
______________for the true population mean.
Stat203
Fall 2011 – Week 7 Lecture 1
Page 9 of 23
Values outside the interval, are ________ to
be the true value. Values inside are more likely
to be the true value.
Stat203
Fall 2011 – Week 7 Lecture 1
Page 10 of 23
Midterm Review
Topics:
 Types of Data
 Descriptive Statistics and Graphical
Summaries
 Probability
 Normal Distribution
 Sampling and the Sampling Distribution
 Confidence Intervals
Stat203
Fall 2011 – Week 7 Lecture 1
Page 11 of 23
Types of Data
Discrete
o Nominal – no natural ordering
eg: blue / green / brown
o Ordinal – an order, but difference
between categories is unclear
eg: agree/somewhat agree/..
o Interval – ordered with meaningful
difference
eg: # students in this class
Continuous
o Interval – ordered with meaningful
difference
eg: salary
o Ratio – measurements to fraction of
a value
eg: weight or height
Things to know:
Stat203
Fall 2011 – Week 7 Lecture 1
Page 12 of 23
-
definitions
questions will be similar to assignment
know how to identify each type of variable
know effective summary measures and
how to interpret graphs of each type
Stat203
Fall 2011 – Week 7 Lecture 1
Page 13 of 23
Descriptive Statistics
Frequency Distribution
Relative Frequency Distribution
Percent Frequency Distribution
Cumulative Frequency Distribution
Percentiles
Ratios
Measures of Central Tendency
- mean, median, mode
Measures of Variability
- range, IQR, standard deviation
Stat203
Fall 2011 – Week 7 Lecture 1
Page 14 of 23
Things to know:
Frequencies:
- as in earlier lectures, and the assignment,
understand how to calculate the
frequency, percent frequency and
cumulative frequency distributions
- understand how to utilize row and column
totals in a cross – tab
- questions may as you to fill in missing
numbers in a table or to interpret percent
frequencies
Measures of Central Tendency:
- understand strengths and weaknesses of
each measure
- understand how to calculate each one
- be able to interpret each
Measures of Variability:
Stat203
Fall 2011 – Week 7 Lecture 1
Page 15 of 23
- have a basic understanding of variability
(ie: understand high compared to low
variability)
- understand how each measure is
calculated
- be able to interpret each measure
Stat203
Fall 2011 – Week 7 Lecture 1
Page 16 of 23
Graphical Presentations
Barplots
Histograms
Pie Charts
Box plots
Things to know:
- be able to interpret each plot
- understand left and right (or positive or
negative) skew from histogram or boxplot
- understand what a symmetric distribution
looks like
- be able to make statements about the
data based on a barplot
Stat203
Fall 2011 – Week 7 Lecture 1
Page 17 of 23
Probability
Know and know how to apply the 4 rules:
- probability between 0 and 1
- probability of converse events (the 1
minus trick)
- addition rule
- multiplication rule
Understand and be able to define an event of
interest.
Know how to interpret a probability distribution
(for discrete and continuous events) and be
able to use it to determine the probability of
certain events.
Stat203
Fall 2011 – Week 7 Lecture 1
Page 18 of 23
Normal Distribution
Properties / Characteristics of a Normal
Distribution
- shape, symmetry, 68-95-99 rule, mean,
standard deviation
Understand what a Standard Normal
distribution is
If given a variable that is Normally distributed,
you should be able to standardize it and / or
calculate the probability of certain events.
eg: length of pregnancy or size of bladder
Stat203
Fall 2011 – Week 7 Lecture 1
Page 19 of 23
Sampling and the Sampling Distribution
Methods of Sampling
- Non Random
o volunteer sampling
o convenience sampling
- Random sampling
o simple random sample
o systematic random sample
o stratified random sample
o cluster (multistage) sample
Statistics versus Parameters
Understand that statistics vary from sample to
sample  sampling distribution
Law of Large Numbers
Central Limit Theorem
Things to know:
Stat203
Fall 2011 – Week 7 Lecture 1
Page 20 of 23
- how to identify each type of sample and
why it might be chosen
- understand the law of large numbers
- if told about a data set, be able identify
the sampling distribution of the mean (ie:
be able to apply the central limit theorem)
- be able to calculate probabilities related to
the mean (eg: the air-conditioner repair
example)
Stat203
Fall 2011 – Week 7 Lecture 1
Page 21 of 23
Confidence Intervals
When  is known and when  is unknown.
How to calculate given summary statistics
(including how to look up the correct t-value)
Interpretation
Stat203
Fall 2011 – Week 7 Lecture 1
Page 22 of 23
Suggestions for Studying and
writing the Exam
Open book - open notes
No Laptops
Calculators are OK, but not absolutely necessary.
Questions will be an assortment of short answer,
fill in the blank, or multiple choice.
Review the summary of topics from each lecture - there will be some easy marks!
Consider the homework as sample problems.
Review examples in my notes
If in doubt about definitions in my notes, or the
text, go with my notes
If you think there’s an error in my notes, email me!
Stat203
Fall 2011 – Week 7 Lecture 1
Page 23 of 23