Stat 217 – Day 12Normal Distribution (Topic 12)

Upcoming work

Lab 4 due Thursday 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

Where we are going

Same issues 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!

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

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)

Relative Frequency over time

50 spins

200 spins

500 spins

1000 spins

-10

-20

-33

-86

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…

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?

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”

Next topic

Calculating probabilities from a “normal probability model”

Is it surprising for a random person to have body temperature above 99.5oF?

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

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

Solution Approach 2

If body temperatures have mean 98.25 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?

70.1733.

25.985.99

z

Empirical rule

Do you suspect body temperatures follow a reasonably symmetric, mound-shaped distribution?

Can we do better?

16%

2.5%

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

3) Mathematical model (p. 234)

Calculating probabilities

1) Table II

See online demo

2) Applet: Normal probability calculator

See online demo

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

Activity 12-2

(a)

(c) z = (2500-3300)/570 = -1.40

(d) Technology: .0802

3300

570

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

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)

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

P. 623

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

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

Exam 1 Comments

Heart attacks and pets 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

Exam 1 Comments

Veterans vs. nonveterans 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!