Stor 155, Section 2, Last Time Review…. Stat 31 Final Exam: Date & Time: Tuesday, May 8,...

Stor 155, Section 2, Last Time• Review…

Stat 31 Final Exam:Date & Time:

Tuesday, May 8, 8:00-11:00

Last Office Hours:• Thursday, May 3, 12:00 - 5:00• Monday, May 7, 10:00 - 5:00 • & by email appointment (earlier)

Bring with you, to exam:• Single (8.5" x 11") sheet of formulas• Front & Back OK

Review Slippery Issues

Major Confusion:

Population Quantities

Vs.

Sample Quantities

Levels of Probability• Simple Events

– Big Rules of Prob (Not, And, Or)– Bayes Rule

• Distributions (in general)– Defined by Tables

• Summary of discrete probs• Get probs by summing

– Uniform• Get probs by finding areas

Levels of Probability• Distributions (in general)

• Named (& Useful) Distributions– Binomial

• Discrete distribution of Counts• Compute with BINOMDIST & Normal Approx.

– Normal• Continuous distribution of Averages• Compute with NORMDIST & NORMINV

– T• Similar to Normal, for estimated s.d.• Compute with TDIST & TINV

Today’s Focus

• Decisions you need to make

• While taking Final Exam

• When faced with a word problem

• Key to deciding on approach

(knowing which formula to use)

Review Decisions NeededMain Challenge: Word problems on

statistical inference

Choices to keep in mind:

1. Big picture:

a. Single Sample

b. Two Samples

c. Two Way Tables

d. Regression

Review Decisions Needed2. Probability model:

a. Proportions – Counts

(p based)

b. Normal Means – Measurements

(mu based)


a. Proportions – Counts (p based)

i. Best Guess

ii. Conservative

iii. BINOMDIST

iv. Normal Approx to Binomial

(used usually for Hypo tests, etc.)


b. Normal Means (mu based)

i. Sigma known – NORMDIST &

NORMINV

ii. Sigma unknown – TDIST & TINV


(Keeping Excel functions straight)

Cutoff → Prob Prob → Cutoff

Counts, Prop’ns BINOMDIST ???

Meas. σ known NORMDIST NORMINV

Meas. σ unkno’n TDIST TINV



• Recall horrible Excel Organizations

• Different functions work differently

• Indicate these on formula sheet…



What about ???:

• There is no BINOMINV

• Since tricky to invert discrete prob’s

• Have to use Normal Approx to Binomial

Review Decisions Needed3. Inference Type:

a. Confidence Interval

b. Choice of Sample Size

c. Hypothesis Testing

(each has its set of formulas…)


a. Confidence Interval

i. Binomial type: Best guess, NORMINV

ii. Binomial type: Conservative, NORMINV

iii. Normal, σ known: NORMINV or CONFIDENCE

iv. Normal, σ unknown: TINV



b. Choice of Sample Size

i. Binomial type: Best guess, NORMINV

ii. Binomial type: Conservative, NORMINV

iii. Normal, σ known: NORMINV

iv. Normal, σ unknown: TINV



c. Hypothesis Testing – P-values

i. Binomial type: NORMDIST (or BINOMDIST)

ii. Normal, σ known: NORMDIST

iii. Normal, σ unknown: TDIST

iv. Variation, σ known: Z-stat

v. Variation, σ unknown: t-stat


Review Decisions NeededSummary of decisions

1. Big picture:

(Single - Two Samples – 2 Way Tab’s – Reg’n)

2. Probability model:

(Prop’ns (Counts) - Normal (Meas’ts))

3. Inference Type:

(Conf. Int. - Sample Size – Hypo Testing)

Practice Making Decisions• Print all HW pages

• Randomly choose page

• Randomly choose problem

• Work that out (make decisions…)

• Mark it off

• Return & repeat

• Finish all correctly?

An easy A in this course

And Now for Something Completely Different

Einstein was once traveling from Princeton on a train when the conductor came down the aisle, punching the tickets of each passenger.

When he came to Einstein, Einstein reached in his vest pocket. He couldn't find his ticket, so he reached in his other pocket.


It wasn't there, so he looked in his briefcase but couldn't find it. Then he looked in the seat by him. He couldn't find it.

The conductor said, "Dr. Einstein, I know who you are. We all know who you are. I'm sure you bought a ticket. Don't worry about it." Einstein nodded appreciatively.


The conductor continued down the aisle punching tickets.

As he was ready to move to the next car, he turned around and saw the great physicist down on his hands and knees looking under his seat for his ticket.


The conductor rushed back and said, "Dr. Einstein, Dr. Einstein, don't worry. I know who you are. No problem. You don't need a ticket. I'm sure you bought one."


Einstein looked at him and said, "Young man, I too know who I am. What I don't know is where I'm going."

A Request> Hi Professor Marron,

>

> For the review session, can we please go over the hypothesis testing and

> when to use the one or two sided tests, and the overall process for

> hypothesis testing? Thanks!!

Response1. In review from April 19, did:

(Hypo Testing: Pop’n vs. Sample)

(So just do quick reminder here)

2. So here focus on 1-sided vs. 2-sided

Hypothesis Testing – Z scores

E.g. Fast Food Menus:

Test

Using

P-value = P{what saw or m.c.| H0 & HA bd’ry}

(guides where to put $21k & $20k)

000,20$:0 H

000,20$: AH

10,400,2$,000,21$ nsX

Hypothesis Testing – Z scores

P-value = P{what saw or or m.c.| H0 & HA bd’ry}

rybdXP '|000,21$

000,20$|000,21$ XP

102400$

000,20$000,21$

nsX

P

317.1 ZP

Response2. So here focus on 1-sided vs. 2-sided

This was studied in detail on March 22,

So review that

But also consider Variations, i.e. how to twiddle problem to get opposite answer

Hypothesis Testing, III

CAUTION: Read problem carefully to distinguish between:

One-sided Hypotheses - like:

Two-sided Hypotheses - like:

:.:0 AHvsH

:.:0 AHvsH

Hypothesis TestingHints:• Use 1-sided when see words like:

– Smaller– Greater– In excess of

• Use 2-sided when see words like:– Equal– Different

• Always write down H0 and HA – Since then easy to label “more conclusive”– And get partial credit….

Hypothesis Testing

E.g. Text book problem 6.34:

In each of the following situations, a

significance test for a population mean,

is called for. State the null hypothesis,

H0 and the alternative hypothesis, HA

in each case….

Hypothesis TestingE.g. 6.34aAn experiment is designed to measure the

effect of a high soy diet on bone density of rats.

Let = average bone density of high soy rats = average bone density of ordinary rats

(since no question of “bigger” or “smaller”)

O

OHSH :0OHSAH :

HS

VariationE.g. 6.34aAn experiment is designed to see if a high

soy diet increases bone density of rats.Let = average bone density of high soy rats = average bone density of ordinary rats

(since no question of “bigger” or “smaller”)

O

OHSH :0OHSAH :

HS

Hypothesis TestingE.g. 6.34bStudent newspaper changed its format. In a

random sample of readers, ask opinions on scale of -2 = “new format much worse”, -1 = “new format somewhat worse”, 0 = “about same”, +1 = “new a somewhat better”, +2 = “new much better”.

Let = average opinion score

Hypothesis TestingE.g. 6.34b (cont.)

No reason to choose one over other, so do two sided.

Note: Use one sided if question is of form: “is the new format better?”

0:0 H

0: AH

Hypothesis TestingE.g. 6.34cThe examinations in a large history class are

scaled after grading so that the mean score is 75. A teaching assistant thinks that his students have a higher average score than the class as a whole. His students can be considered as a sample from the population of all students he might teach, so he compares their score with 75.

= average score for all students of this TA75:0 H 75: AH

VariationE.g. 6.34cThe examinations in a large history class are

scaled after grading so that the mean score is 75. A teaching assistant thinks that his students have a different average score from the class as a whole. His students can be considered as a sample from the population of all students he might teach, so he compares their score with 75.

= average score for all students of this TA75:0 H 75: AH

Hypothesis Testing

E.g. Textbook problem 6.36

Translate each of the following research

questions into appropriate and

Be sure to identify the parameters in each

hypothesis (generally useful, so already

did this above).

0H AH

Hypothesis TestingE.g. 6.36aA researcher randomly divides 6-th graders

into 2 groups for PE Class, and teached volleyball skills to both. She encourages Group A, but acts cool towards Group B. She hopes that encouragement will result in a higher mean test for group A.

Let = mean test score for Group A = mean test score for Group BAB

Hypothesis TestingE.g. 6.36a

Recall: Set up point to be proven as HA

BAH :0

BAAH :

VariationE.g. 6.36aA researcher randomly divides 6-th graders

into 2 groups for PE Class, and teached volleyball skills to both. She encourages Group A, but acts cool towards Group B. She wonders whether encouragement will result in a different mean test for group A.

Let = mean test score for Group A = mean test score for Group BAB

VariationE.g. 6.36a

Recall: Set up point to be proven as HA

BAH :0

BAAH :

Hypothesis TestingE.g. 6.36bResearcher believes there is a positive

correlation between GPA and esteem for students. To test this, she gathers GPA and esteem score data at a university.

Let = correlation between GPS & esteem

0:0 H

0: AH

VariationE.g. 6.36bResearcher investigates the potential

correlation between GPA and esteem for students. To test this, she gathers GPA and esteem score data at a university.

Let = correlation between GPS & esteem

0:0 H

0: AH

Hypothesis TestingE.g. 6.36cA sociologist asks a sample of students

which subject they like best. She suspects a higher percentage of females, than males, will name English.

Let: = prop’n of Females preferring English = prop’n of Males preferring English

Fp

MF ppH :0

MFA ppH :

Mp

VariationE.g. 6.36cA sociologist asks a sample of students

which subject they like best. Is there a difference between the percentage of females & males, that name English.

Let: = prop’n of Females preferring English = prop’n of Males preferring English

Fp

MF ppH :0

MFA ppH :

Mp

Stor 155, Section 2, Last Time Review…. Stat 31 Final Exam: Date & Time: Tuesday, May 8,...

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