1. Statistics 200 Objectives:
Lecture #20 Thursday, October 27, 2016
Textbook: Sections 9.6, 11.1, 11.2
Objectives:
• Apply sampling distribution for one sample mean to confidence intervals and hypothesis tests.
• Identify situations in which t-multipliers and t-tests should be used instead of z-multipliers and z-tests.

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. Categorical data (2 categories) Quantitative data
parameter:
statistic:
parameter:
statistic:

4. Clicker Question: Consider the following three survey questions:
Do you plan to vote in the upcoming presidential election?
How old are you?
Which candidate do you most support?
How many of these questions will produce Quantitative data? 1 2 3

5. Example 1: The population is normally distributed.

6. Clicker Question
Which statement(s) are false, when comparing the original distribution to the two sampling distributions
All three distributions have the same value for the mean
As the sample size increases, the standard deviation for the sampling distribution decreases
The original distribution will always have a smaller standard deviation than what is found with either of the two sampling distributions
The sampling distributions suggest possible values for the population mean.

7. Example 1: The population is normally distributed.

8. Example 1: The population is normally distributed.

9. Sampling Distribution: Standard Deviation & Standard Error
Statistic
Standard Deviation
We generally do not know p.
Thus, we don’t know s.d.(p-hat).
Similarly: We generally do not know σ.
Thus, we don’t know s.d.(x-bar).

10. Sampling Distribution: Standard Deviation & Standard Error
Statistic
Standard Deviation
Standard Error:
Estimated St Dev
if p is unknown, use:
Here, s is the sample standard deviation.
if σ is unknown, use:

11. Example 1: The population is normally distributed.
Substitute s for σ:

12. Standard normal (dotted red) vs. t (solid black)
Degrees of freedom for t distribution:
1, 5, and 20
(as d.f. increases, the t looks more like the standard normal.)

13. Standard normal (dotted red) vs. t (solid black)
Degrees of freedom for t distribution:
1, 5, and 20
We worry a lot about teaching t vs. z, but the difference is tiny for degrees of freedom usually seen in practice.

14. Confidence Interval Formula
sample estimate ± (margin of error)
sample estimate ± (multiplier × standard error)
Generic
Formula:
Specific for
Population Mean: µ
This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996]
This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures.
References:
Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005).
Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50
Here, t* depends on confidence level and df = (n – 1).

15. Multipliers: from the t table (not a complete list, obviously)
Conf. level:
0.90
0.95
0.98
0.99
1 df
6.31
12.71
31.82
63.66
2 df
2.92
4.30
6.96
9.92
3 df
2.35
3.18
4.54
5.84
9 df
1.83
2.26
2.82
3.25
20 df
1.72
2.09
2.53
2.85
30 df
1.70
2.04
2.46
2.75
Infinite df
1.645
1.96
2.326
2.576

16. Clicker Question: What kind of variable is this?
Categorical
Quantitative
Example 2:
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?

17. 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
This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996]
This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures.
References:
Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005).
Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50
Thus, the 95% CI is

18. 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)
This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996]
This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures.
References:
Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005).
Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50

19. 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.
This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996]
This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures.
References:
Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005).
Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50

20. Example 3: The population is NOT normally distributed.

21. 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

22. If you understand today’s lecture…
9.61, 9.62, 9.64, 9.65, 11.25, 11.30, 11.32, 11.33
Objectives:
• Apply sampling distribution for one sample mean to confidence intervals and hypothesis tests.
• Identify situations in which t-multipliers and t-tests should be used instead of z-multipliers and z-tests.