Download Four possible outcomes of a hypothesis test

Four possible outcomes of a hypothesis test
Reality
Decision
Retain H0
H0 true
Correct
Decision
H0 false
Miss:
Type II Error
β
1-α
Reject H0
False Alarm:
Type I Error
α
Correct
Decision
1 – β (power)
Confidence Intervals
A range of values that, with a known degree
of certainty, includes an unknown population
characteristic, such as a population mean
Confidence Intervals:
Estimating a Population Mean, μ
•  Want to estimate average age of college students here
•  Don’t want to sample EVERY STUDENT
•  Instead, I could take a sample (e.g., this class) & estimate μ
•  Xbar! This is called a “point estimate”
• 
• 
The purpose of a CI is to give an “interval estimate” of the
population mean, μ with some degree of certainty
e.g., “We are 95% confident that the mean age of all
students is between 18-24 years old”
Confidence Intervals
•  Used when we collect a sample from a larger population
•  Has two parts
–  A range of number values (e.g., 18-24)
–  A confidence level (e.g., 95%)
•  There is a trade-off between confidence and the size of
your confidence interval
–  If I want to be 100% confident of my range would be very large
–  If I want to be 10% confident, my range can be very small
•  CI’s do not mean that there is a 95% chance that the CI
includes the true population mean
–  CI’s either do or don’t include the population mean…
–  “I am 95% confident that my CI includes the true population mean”
–  “If I ran this study 100 times and made 100 confidence intervals, 95
of my 100 CI’s would include the true population mean”
These
confidence
intervals don’t
contain the true
value of the
mean L
X-bar…aka the
sample mean
This means
you add AND
subtract!
CI = X ± z *(σ X )
Sigma x-bar
(lower case!) …
aka the standard
error or σ/√n
critical value of z
€
Let’s calculate!
This means
you add AND
subtract!
X-bar…aka the
sample mean
CI = X ± z *(σ X )
Sigma x-bar
(lower case!) …
aka the standard
error or σ/√n
critical value of z
•  Does the mean GPA of students who take statistics differ from the
mean GPA of all college students?
–  μ = 2.7, σ = 0.6, X-bar = 2.85, n = 36
€
•  Set critical values at 1.96
•  Calculate:
•  Upper CI:
•  Lower CI:
X-bar…aka the
sample mean
Let’s calculate!
This means
you add AND
subtract!
CI = X ± z *(σ X )
Sigma x-bar
(lower case!) …
aka the standard
error or σ/√n
critical value of z
•  Does the mean GPA of students who take statistics differ from the
mean GPA of all college students?
–  μ = 2.7, σ = 0.6, X-bar = 2.85, n = 36
€
•  Set critical values at 1.96
•  Calculate: σ x =
.6
= 0.1
36
•  Upper CI: 2.85 + 1.96 * .1 = 2.654
•  Lower CI: 2.85 - 1.96 * .1 = 3.046
I am 95% confident that the mean GPA of students who takes statistics falls
between 2.65 and 3.05.
So what?
•  The confidence interval we constructed based on our
sample includes the overall population mean (2.7)
•  So this is just another way of saying that this sample and
our population mean are not significantly different
•  That is, we can’t reject our null hypothesis (H0)
2.70
2.654
2.85
3.046
Practice!
A sample of 64 high school athletes from Chicago are
evaluated by their coaches to determine whether they should
be encouraged to try out for college sports teams. A score of
4 indicates an acceptable level of athletic ability. A score less
than 4 indicates a poor level of athletic ability. A score greater
than 4 indicates that they are a superb athlete.
The mean score for the sample is 3.82. The population
standard deviation is 0.4
•  Construct a 95% confidence interval for the unknown population
mean (don’t forget to convert the standard deviation to standard
error!)
•  Interpret the confidence interval. Do you think there is evidence
for the athletes being really awesome or really terrible?
Oh so many ways to calculate a
confidence interval
z test CI: CI = X ± z *(σ X )
Single Sample t CI: CI = X ± t *(sX )
Dependent t CI:
CI = D ± t *(sD )
€
Independent t CI: €
CI = (X1 − X 2 ) ± t * (sX1 −X 2 )