Download Confidence Intervals: Estimating a Population Mean, μ

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Student's t-test wikipedia, lookup

German tank problem wikipedia, lookup

Regression toward the mean wikipedia, lookup

Misuse of statistics wikipedia, lookup

Confidence Intervals:
Estimating a Population Mean, μ
Want to estimate average age of San Diego college students
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
San Diego college students is between 18-24 years old”
Confidence Intervals
Used when we can only collect a sample from a large
If we had data from the entire population, we could say
EXACTLY what the mean value is with 100% confidence
A Confidence Interval (CI) has two parts:
Size: range of number values (i.e., 7.3-8.4)
Confidence Level (i.e., 95%)
Trade-off between confidence and size of confidence
If I want to be 100% confident in my estimate, would my
range be very small or very large?
If I only need to be 10% confident, my range can be small
What do CI’s Mean?
CI  X  z *( X )
 •
What confidence intervals DO NOT mean:
“There is a 95% chance that my CI includes the true
population mean” - NOT TRUE!
Either it DOES or it DOESN’T!
“I am 95% confident that my CI includes the
true population mean”
Even better: “If I ran this study 100 times, and
made 100 CI’s in this manner, 95 of my 100
CI’s would include the true population mean”
Confidence Interval
z test example: Does the mean GPA of
students who take statistics differ from the
mean GPA of all college students? Construct
a 95% confidence interval, given: μ = 2.7, σ =
0.6, Xbar = 2.85, sample size n = 36.
σXbar = 0.6/√36 = 0.1
μ = 2.7
z* = -1.96
z* = +1.96
Confidence Intervals
CI  X  z *( X )
Xbar = 2.85
z = ±1.96
σXbar = 0.6/√36 = 0.1
–Xbar = 2.85 ± 1.96*(0.1)
–Xbar = 2.85 ± 0.196
–Xbar = between 2.654 and 3.046
So What???
“We are 95% confident that the population mean lies
between 2.654 and 3.046.”
The confidence interval we constructed based on our
sample includes the overall population mean: 2.70
This is another way of saying that this sample and
our population mean aren’t significantly different!
That is, we shouldn’t reject H0
σXbar = 0.1
Confidence Interval Formulas
z test CI:
CI  X  z *( X )
Single Sample t CI:
CI  X  t *(sX )
Dependent t CI:
Independent t CI:
CI  D  t *(sD )
CI  (X1  X 2 )  t * (sX1 X 2 )
Ch. 12 hw
A sample of 25 statistics students eats a mean of 9 candy bars per month; it is
known that the population mean for college students is 8, with a standard
deviation of 4.
1a. Construct a 95% confidence interval for the true population mean of statistics
students’ candy bar eating.
b. Does this CI indicate a significant difference between statistics students and
other students (why or why not)?
c. Compare & contrast this analysis with a 2-tailed z-test on the same data.
2. Say a different sample from another college yielded a 95% confidence interval of
8.5 ± 1.5. True or false:
a. The true population mean is between 7 and 10.
b. About 95% of these students sampled scored between 7 and 10.
c. There is a 95% chance that this CI contains the true population mean.
d. 5% of confidence intervals constructed this way will not contain the true
population mean.
3. The college candy salesman wants a more precise confidence interval to better
estimate inventory. Name two ways to obtain a more precise CI.