1. Daniela Stan Raicu School of CTI, DePaul University
CSC 323 Quarter: Winter 02/03
Daniela Stan Raicu
School of CTI, DePaul University
11/18/2018
Daniela Stan - CSC323

2. Outline Chapters 6 & 7 & 8: Confidence Intervals
Review on Chapter 6 – confidence intervals for population average
Review on Chapter 8 – confidence intervals for population proportion
Chapter 7:
Confidence intervals when the population distribution is unknown
11/18/2018
Daniela Stan - CSC323

3. Is x normal distributed?
Is the population normal?
Yes No
Is ?
Is ?
Yes No
Yes No
is normal
has
t-student
distribution
is considered
to be normal
may or
may not be
considered normal
(We need more info)
11/18/2018
Daniela Stan - CSC323

4. Assumptions when applying z-statistic
1. The population has a normal distribution with mean µ and standard deviation .
2. The standard deviation  is known
3. The size ‘n’ of the simple random sample (SRS) is large
4. The appropriate test statistic to use for inference about µ when  is known
is the z statistic:
where the expected value µ0 is the value assumed in the null hypothesis Ho.
z has a normal distribution N(0,1)
z =
(x - µ0)
 /  n
11/18/2018
Daniela Stan - CSC323

5. Assumptions when applying z-statistic
Is z-statistic appropriate to use when:
The sample size is small?
2. The population does not have a normal distribution?
3. The population has a normal distribution but the standard deviation  is unknown?
When the standard deviation of a statistic (in our case x) is estimated from data, the result is called the standard error of the statistic:
SE x = s/  n
What is the distribution of
(x - µ0)
s/  n ?
It is not normal!
11/18/2018
Daniela Stan - CSC323

6. Inference on averages for small samples (cont.)
If data arise from a population with normal distribution and n is small (n<30), we can use a different curve, called t- distribution or Student’s curve.
The t-distribution was discovered by W. S. Gosset (born on 13 June 1876
in Canterbury, England), the chief statistician of the Guinness brewery in
Dublin, Ireland. He discovered the t-distribution in order to deal with small
samples arising in statistical quality control. The brewery had a policy against
employees publishing under their own names, thus he published his results
about the t-distribution under the pen name "Student", and that name has
become attached to the distribution.
11/18/2018
Daniela Stan - CSC323

7. The t-student distributions
Suppose that an SRS of size n is drawn from an N(µ, ). Then the one-sample t statistic
t =
(x -µ0)
s /  n
has the t-distribution with n-1 degrees of freedom.
- The degrees of freedom come from the standard deviation s in the denominator of t.
- There are many student’s curves! There is one student’s curve for each number of degrees of freedom; for tests on averages:
Degrees of freedom = number of observations – 1
11/18/2018
Daniela Stan - CSC323

8. Comparing the student’s curve and the standard normal curve
d.f.=5
d.f.=15 t t
Student’s curve
Standard Normal curve
Student’s curve has “fatter” tails. For d.f. around 30, the student’s curve is very similar to the standard normal curve.
d.f.=30
11/18/2018
Daniela Stan - CSC323 t

9. When to use the t-test
When should we use it? Each of the following conditions should hold:
For computing a statistical test on averages.
The sample is a simple random sample.
The number of observations is small, the sample size n is less than 30.
The distribution of the population is bell-shaped, it is not too different from the normal distribution. (Not easy to check, typically true for measurements!)
11/18/2018
Daniela Stan - CSC323

10. Tests on averages: z-test or t-test?
If the amount of current data is
large
Small (n <30)
Use the z-test & the normal curve
The distribution of the population is
Unknown but quite different from the normal curve
Unknown but not different from the normal curve
Use the t-test & the student’s curve
Do not use the t-test!
11/18/2018
Daniela Stan - CSC323

11. Confidence intervals for proportions
Assignment:
1. Draw the flowchart for estimating the population
proportions
2. Calculate the confidence interval for the population proportion
for different situations from the flowchart
3. Calculate the confidence intervals for different confidence
levels such as C=.96, .98, etc.
4. Give examples where the way we calculated the confidence
Intervals does not work.
Review assignment #5
11/18/2018
Daniela Stan - CSC323