1. Statistics 200 Objectives:
Lecture #21 Tuesday, November 1, 2016
Textbook: 9.7, 9.8, 11.1, 11.2, 11.3, 11.4
Objectives:
• Apply sampling distribution for one sample mean to confidence intervals.
• Apply sampling distribution for difference of two sample means to confidence intervals.
• Apply sampling distribution for sample mean of (paired) differences to confidence intervals.
• Recognize similarities between one mean and mean of paired differences.

2. We have begun a strong focus on Inference
Means
Proportions
One population mean
One population proportion
Two population proportions
Difference between Means
Mean difference
This week

3. Example 2 from Thursday:
Clicker Question: What kind of variable is this?
Categorical
Quantitative
Example 2 from Thursday:
We ask each of 31 students “how many regular ‘text’ friends do you have?”
Survey results:
n = X-bar = 6 friends s = 2.0 friends
Calculate a 95% Confidence Interval:
How can we estimate the population mean number of regular “text” friends for all STAT 200 students using these data?

4. Confidence Interval Formula
sample estimate ± (margin of error)
sample estimate ± (multiplier × standard error)
Generic
Formula:
Survey results:
n = X-bar = 6 friends s = 2.0 friends
Thus, the 95% CI is

5. Confidence Interval Interpretation
We are 95% confident that the…
sample mean
sample proportion
population mean
population proportion
range of values for the
…number of regular “text” friends for STAT 200 students is between 5.3 and 6.7 friends.
Calculated Interval:
6.0 ± 0.7 friends
(5.3 to 6.7 friends)

6. Confidence Interval Conclusion
95% C.I.: to 6.7 friends
In the population, we may conclude, with 95% confidence, that on average, STAT 200 students have
A. more than 6 friends.
more than 4 friends.
fewer than 5 friends.
fewer than 6 friends.

7. Are all sampling distributions normal? _____
When do we have to be cautious?
with _____ sample sizes
where the original population is not ______ in shape No
small
normal
One-Sample t procedure is valid
if one of the conditions for normality is met:
Sample data suggest a normal shape
We have a large sample size (n ≥ __) or 30
Sampling distribution will look normal in shape

8. Parameter of interest:
Example: Compare predicted GPAs of males and females in STAT 200
Q: What do you think your actual GPA will be when you gradaute?
Students in this class: Representative sample (?) of all STAT 200 students, with nf=157 and nm=130.
Parameter of interest:

9. • Parameter of interest: • Estimate of the parameter:
Example: Compare predicted GPAs of males and females in STAT 200
• Parameter of interest:
• Estimate of the parameter:
• Statistics collected from the sample:
What do we need in order to create a CI for ?

10. sample estimate ± (multiplier × standard error)
Formula for CI for
sample estimate ± (multiplier × standard error)
Roughly 2.0 for
95% confidence
???
What is the standard error of ?

11. Example: Compare predicted GPAs of males and females in STAT 200
Here are the data, summarized:
Thus,
• Estimate = – 3.456, which is
• Multiplier = roughly 2 (more on this later…)
• SE (estimate) =

12. Example: Are smokers and non-smokers different heights on average?
Summary of class data from Minitab:
Variable SmokeCig N N* Mean SE Mean StDev
Height No
Yes
Based on these data, a 95% CI would be roughly:
264 – 19 ± 2 × sqrt( )
67.28 – ± 2 × sqrt( )
264 – 19 ± 2 × sqrt( )
67.28 – ± 2 × sqrt( )
However, there is a slight problem with the multiplier of 2…

13. Clicker Question: How many of those statements are TRUE?
Example: Computer versus TV
25 students in a liberal arts course were given a survey that asked them how many hours per week they watched television and how many hours per week they used a computer.
The goal is to determine if there is a difference in the mean number of hours spent per week on computers versus TV.
Consider the statements below:
The two samples are dependent
The experimental unit is a student
The response variable is quantitative
This is a randomized experiment
Clicker Question: How many of those statements are TRUE?
0 C. 2
1 D. 3

14. Example: Computer versus TV
(experimental) unit:
student
student
Computer TV 1 30 20 2 25 3 10 4 5 “
15.0
response variable:
Number of hours
Variation you want to…
reduce: the variation from student to student
explain: the variation due to type of screen

15. What if we construct a independent samples CI?
Difference = mu (Computer) - mu (TV)
95% CI for difference: (-3.29, 9.17)
Conclusion:
Since the C.I. contains ____, can ______ claim that a difference exists.
not

16. dependent two Paired t Problem:
The two samples are ___________ (paired)
______ measurements on each unit
When the two-sample t procedure is incorrectly used,
it captures unwanted variation found with the two individual standard deviations
It is less able to find significance
Instead use: _________ procedure
dependent
two
Paired t

17. Data used for paired analysis
What do you notice
when examining the
signs of the differences?

19. Summary Statistics for Samples
student
Computer TV
Comp - TV 1 30 20 10 2 25 -5 3 4 5 …
15.0
Mean
17.04
14.10
2.94
StDev
s1 =12.36
s2 = 9.26
sd = 5.34
Mean of the differences

20. Confidence Interval (key: treat it like a single mean)
Calculate a 95% confidence interval to estimate the population mean difference in hours spent on a computer vs watching TV
5.34
2.9
n = 25 students 25
df = n-1 = 24
Calculation:
2.9 ± [2.06 × 5.34/sqrt(25)] = 2.9 ± 2.2 =
0.7 to 5.1

21. If you understand today’s lecture…
11.25, 11.30, 11.32, 11.33, 11.45, 11.46, 11.52, 11.53, 11.55
Objectives:
• Apply sampling distribution for one sample mean to confidence intervals.
• Apply sampling distribution for difference of two sample means to confidence intervals.
• Apply sampling distribution for sample mean of (paired) differences to confidence intervals.
• Recognize similarities between one mean and mean of paired differences.