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Hypothesis Testing 2

Lecture Goals

You should be able to:

- Apply the principles of hypotheses testing to samples where n > 1
- Understand the relationship between sampling error and hypothesis testing when n > 1
- Understand decision errors
- Understand the relationship between errors and alpha Reading

Chapter 12 pgs. 267-282

Definitions

hypothesistesting = a tool for justifying generalizations about population based on sample data

Usually sample size > 1

A researcher wants to know whether the mean IQ score of PSU students is different from the mean IQ score of all Americans.

The IQ scores of all Americans are normally distributed with a m = 100 and s = 16.

The researcher randomly selects a sample of 10 PSU students and finds that their mean IQ score is 111.

Same Idea As Before

Did an unlikely event occur? Q = Are the numbers from the same pop. of scores or different pop. of scores?

Q = How likely is it that the scores (means) are from the same pop. of scores?

A = Depends on sampling error & criterion that an unlikely event has occurred

Sampling Error Random sample = each member of the pop. has equal opportunity to be selected

= should be representative of the population (x should = m )

Some random samples are more representative of the pop. than others

101 people take a test. The scores are below.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
88 89 90 91 92 93 94 95 96 97 98 99 100

Median = 50

Mean = 50

Now...we select a random sample of 5 scores from this population of scores.

random sample 1: 38 40 50 67 88

Mean = 56.6 Median = 50

random sample 2: 0 1 2 3 4

Mean = 2 Median = 2

Chance determines how representative sample is of population

Sampling Error = how far a random sample is from being a representative sample

If account for ALL possible random samples, sampling errors are normally distributed

If you flip 4 fair coins there are 16 possible outcomes

50% chance of H & 50% chance of T

Expect = 2 H and 2 T

Sampling Error = excess heads

Outcome Excess Heads

HHHH 2

HHHT 1

HHTH 1

HTHH 1

THHH 1 Sampling Errors

HHTT 0

HTHT 0 (graph)

THHT 0

HTTH 0

THTH 0

TTHH 0

TTTH -1

TTHT -1

THTT -1

HTTT -1

TTTT -2

Person         Score
A                     1
B                     2
C                     3

m = 1 + 2 + 3 = 6 = 2
3 3

X_AB = 1 + 2 = 3 = 1.5
2 2

Sampling error and random samples

- with replacement
- n = 2

Sample X m Sampling Error

AA 1 2 - 1

AB 1.5 2 -.5

AC 2 2 0

BA 1.5 2 -.5

BB 2 2 0

BC 2.5 2 +.5

CA 2 2 0

CB 2.5 2 +.5

CC 3 2 +1

Sample means are normally distributed

(graph)

Because x expected to = m sampling errors are also normally distributed

(graph)

Sampling Errors are normally distributed
- smaller errors more likely than larger

errors (and expected if x = m ) - the larger the error the more unlikely it is

- large sampling errors = unlikely events

Z score and Sampling Error

Z- score = measure of sampling error Large z score = unlikely event

Now: z = X-m= measure of sampling error
s_{sample means}

(formulas)

Same 7 Step Process As Before State null hypothesis

State alternative hypothesis

Set level of significance

* Calculate the z-score *

Find percentile

Decision about the null hypothesis

Draw a conclusion

A researcher wants to know whether the mean IQ score of PSU students is different from the mean IQ score of all Americans.

The IQ scores of all Americans are normally distributed with a m = 100 and s = 16.

The researcher randomly selects a sample of 10 PSU students and finds that their mean IQ score is 111.

a = .05

Step 1: State the Null Hypothesis

H_{o :}_mpsu = 100 (or m_{psu
= m Amer})

Step 2: State the Alternative Hypothesis

H_{A :}_mpsu ¹ 100 (or m_psu
¹m
Amer) Step 3: Set Level of Significance

a = .05

Step 4: Calculate Z-Score

z = 2.17
(see formula)

Step 5: Find Percentile:

Collum C: Area for Z_2.17 = .0150 Step 6: Decision About H_o Hypothesis is NON-directional a /2 = .05/2 = .025

p = .0150

If p > a /2 fail to reject H_o

Step 7: Draw a Conclusion

The data do not show that the mean

IQ score of PSU students is different

than the mean IQ score of all

Americans

Effects of Sample Size

A researcher wants to know whether the mean IQ score of PSU students is different from the mean IQ score of all Americans.

The IQ scores of all Americans are normally distributed with a m = 100 and s = 16.

The researcher randomly selects a sample of 10 PSU students and finds that their mean IQ score is 120.

Step 4: Calculate Z-Score

z = 3.95

(see formula)
Step 5: Find Percentile: Collum C: Area for Z_3.95 = .0000

Step 6: Decision About H_o

Hypothesis is NON-directional a /2 = .05/2 = .025

p = .0026

If p < a /2 reject H_o

Alternative 7 Step Process State null hypothesis

State alternative hypothesis

Set level of significance

Determine critical z-score & region

Calculate the z-score

Compare to make decision about H_o

Draw a conclusion

A researcher wants to know if the reading proficiency of high school seniors in State College is lower than the national norm. A sample of 100 randomly selected seniors has a mean reading score of 72. The mean of the population is 75 and the S.D. is 16.

a = .05

Step 1: State the Null Hypothesis

H_{o :}_mSC = 75 (m_{SC
= m POP}) Step 2: State the Alternative Hypothesis H_{A :}_mSC < 75 (or m_{SC
< m POP}) Step 3: Set Level of Significance

a = .05

Step 4: Determine critical z-score & region

From collum C a = .05: Z = 1.645

Because negative tail Z = -1.645

(graph) Step 5: Calculate Z-Score

z = -1.88
(see formula)

Step 6: Decision About H_o

z = -1.88 = in critical region
(graph)

Step 6: Decision About H_o

If |Z_calc| ³ |Z_crit| then Reject H_o
If |Z_calc| < |Z_crit| then Fail to Reject H_o

Here: 1.88 > 1.645 so Reject H_o

Step 7: Draw a Conclusion

The data show that the mean score of SC seniors is below the national average

JUDGEMENT ERRORS

H_oActually

True False

Reject Type I no error
error
Decision
about H_o

Fail to R. no error Type II
(don't reject) error

Why Type I Errors

From Before:

Q = Are the results likely if H_o is correct?

A = No

Why did we get these results?

1) an unlikely event occurred
2) our initial assumption is wrong

Odds of # 1 = .0001, # 2 more probable

A = No

Odds of # 1 = .0001, # 2 more probable

Odds of # 1 > 0....an unlikely event MIGHT have occurred and H_o is correct

Why Type II Errors

Idea = look for trends...

Population = 100 people.....90 show improvement with a drug, 10 don't

Could randomly select the 10 and conclude no effect, trend = effect

a and Errors

Q - How do we know how unlikely is unlikely enough to reject H_O
Q - How large does a z score have to be to be considered unlikely?

Answer = a

a = criterion for rejecting H_o

Stricter the crit. = less likely reject any H_o
(true ones or false ones)

Smaller a = stricter the criterion
Smaller a = less likely reject any H_o

Type I error = we reject an H_o that is true
Smaller a = less likely make Type I error

Type II error = we do NOT reject an H_othat is false
Smaller a = more likely make Type II error

Consequences of Errors Type I = reject H_o that is true

Claim we found an effect

(Cure for cancer)

We really didn't

Harmful applications

(use drug as ONLY treatment)

Impede scientific progress

(stop looking)

Type II = believe H_o that is false

Dismiss a real effect

(Cure for cancer)

We don't use it

Miss opportunity to use it

Waste time and $ looking elsewhere

Abandon good idea

Avoiding Errors

Replication

Extension

Skepticism

Determination

Homework - Chapter 10: Problems 3,6,10

Chapter 12: Problems 20-22, 24

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