1. Normal Distribution (Topic 12)
Stat 217 – Day 12
Normal Distribution (Topic 12)

2. Upcoming work Lab 4 due Thursday HW 4 posted soon, due next Tuesday
With partner but individual pre-lab
HW 4 posted soon, due next Tuesday
Exam 1 discussion at end of class
“Course Avg” updated in Blackboard
Warnings…
Syllabus reminder…
Lab 3 Grading

3. Where we are going Same issues More formal inference procedures
How do we collect data
How do we analyze data
How do we make statements about statistical significance, generalizability, causation
More formal inference procedures
Don’t lose “reasoning” of significance!
Good time to review “advice for doing well in course” in syllabus…
Maintain the momentum!

4. Lab 4: Probability
Def: The probability (aka likelihood, chance, odds) of a random event occurring is the long-run proportion (or relative frequency) of times the event would occur if the random process were repeated over and over under identical conditions.
Empirical estimate – simulate the process many times and calculate the proportion of times an event (e.g., no moms get correct baby) occurs
 Has to be a random (repeatable) process

5. Probability Notes
In Roulette, the probability you lose a color bet is .526, so why are casinos such a “big business”?
It’s the proportion (relative frequency) that converges, not the frequency (count)

6. Relative Frequency over time
50 spins
200 spins
500 spins
1000 spins
-10
-20
-33
-86

7. Probability Notes Assuming we have a random, repeatable process
What is the probability of the Saints winning the Super Bowl this weekend?
Calculation vs. evaluation vs. interpretation
The probability of landing heads is .50
I consider this a large or a small probability…
If I were to repeatedly toss a coin, then in the long run 50% of the tosses will land heads…

8. Example
Lab 1: Is it surprising to get 14 or more successes in 16 trials
if no preference?
Lab 2: Is it surprising to a difference in conditional proportions of .044 or more
if no yawning effect?

9. Examples cont. So want to start making formal probability statements
Also notice that these distributions have some common features!
Distributions that are mound-shaped and symmetric with “short tails” are often well modeled by the “normal distribution”

10. Next topic Calculating probabilities from a “normal probability model”
Is it surprising for a random person to have body temperature above 99.5oF?

11. Solution approach 1
Body temperatures: Is it surprising to have a body temperature above 99.50F?
1) How often does a healthy adult have such a temperature?
4 of 130 healthy adults, .031

12. Solution Approach 2
Body temperatures: Is it surprising to have a body temperature above 99.50F?
2) Is it more than 2 standard deviations away?
Standardize the observation:
z = observation-mean standard dev

13. Solution Approach 2
If body temperatures have mean and SD .733, what is the z-score for 99.5?
Can we say more?
Do you suspect body temperatures follow a reasonably symmetric, mound-shaped distribution?

14. Empirical rule Can we do better?
Do you suspect body temperatures follow a reasonably symmetric, mound-shaped distribution?
Can we do better?
16%
2.5%

15. 3) Mathematical model (p. 234)
Do you suspect these data are reasonably modeled by a “normal” distribution?
Calculate probabilities by finding the area under the curve in the region of interest

16. Calculating probabilities
1) Table II See online demo 2) Applet: Normal probability calculator

17. Using technology Normal Probability Calculator applet
Interpretation: If repeatedly sample healthy adults, about 4.4% of them will have a temperature of 99.5 or more

18. Activity 12-2
(a) (c) z = ( )/570 = (d) Technology: .0802
570
3300

19. Interpretation
The probability of a randomly selected baby having “low birth weight” (weight < 2500)  .08
If repeatedly select babies, in the long run will obtain a low birth weight baby about 8% of the time
Approximately 8% of all babies are low birth weight
About 8% of area under the curve is to the left of 2500

20. To do for Tuesday Finish Activity 12-2 using technology
For TIA credit, submit answers to Activity 12-6 (sketches, method)
As come into class, ready to discuss
See also Activity 12-4 (self-check)

21. Converting z-scores to probabilities
Using Table II to find the proportion of the distribution to the left of this z-value…
1. Use first two digits to locate the row
2. Use the hundredths place to locate the column
3. Reports the area to the left of the z-score

22. P. 623

23. Converting z-scores to probabilities
Using Table II to find the proportion of the distribution to the left of this z-value…
1. Use first two digits to locate the row
2. Use the hundredths place to locate the column
3. Reports the area to the left of the z-score
Pr(Z < z)
Pr(body temp < 97.5) = Pr(Z < -1.03) = .1515

24. Exam Comments
Average  .80, full solutions in Blackboard under Course Materials
Infant Sleep Study (from self-check activities)
Conditional distribution
Web user addictions
Parameter = proportion of all internet users who (admit) are addicted
Sampling vs. nonsampling bias

25. Exam 1 Comments Heart attacks and pets Number of close friends
Something else different between those with pets and those without that might explain why those with pets more likely to survive 5 years.
Wealthier?
Number of close friends
Frequency table (Act 9-3, 8-7)
Median position vs. value, make sure makes sense in context!
Skewness affects mean vs. median even without outliers

26. Exam 1 Comments Veterans vs. nonveterans Two-way table Extra credit
Can’t consider only counts when have unequal group sizes!
Conjectured direction vs. statistical significance
Two-way table
p-value interpretation vs. evaluation
Continue to focus on and improve interpretations
Extra credit
Sample size doesn’t help non/sampling bias!