MedI 7 Hypothesis Testing

1

Developing and testing a hypothesis

2

Learning material

Chapters 9-11 Confidence Interval Estimation Chapters 12-14 Testing hypotheses

3

Learning Objectives bull Outline sources of ideashypotheses bull Understand the process of hypothesis testing

bull Understand the purpose and interpretation of confidence intervals

4

Ideas and evidence

Hypotheses (Proposed answers)

Case reports

Biology

Epidemiology Clinical

observation

Imagination

Reasoning

Observational studies

Experimental studies

Tests of hypothesis

EVIDENCE-BASED INFORMATION

IDEAS

5

Examples Mechanisms of disease at the molecular level Drugs against antibiotic-resistant bacteria are developed through

knowledge of the mechanism of resistance

Case reports-Clinical observation Tamoxifen developed for contraception was found to prevent

breast cancer in high-risk women

Beliefs about herbal remedies Aspirin is a naturally occuring substance that has become

established as orthodox medicine after rigorous testing

Epidemiologic studies of populations Observations of low prevalence of colonic diseases (irritable

bowel syndrome appendicitis colorectal cancer) in Africa (where diet is high in fiber) led to efforts to prevent bowel disease with high-fiber diets Comparisons across regions have suggested the value of fluoride to prevent dental caries

6

Learning Objectives

bull Outline sources of ideashypotheses

bull Understand the process of hypothesis testing


7

When to apply Statistical Hypothesis Testing - To answer a research question

Example

Is new antibiotic X a more effective drug for treating

infections than standard treatment Y

To test the folowing hypothesis proportion of survivors in population of patients treated by X= =proportion of survivors in population of patients treated by Y

8

Important terms in Hypothesis Testing

bull Null and Alternative Hypotheses bull P-value bull Type I and Type II Errors bull Significance level bull Confidence level bull Power of a test bull Test statistic bull One(two)-tailed test bull Statistical tests

9

Null and Alternative Hypotheses The null hypothesis (H0) is usually of the form there is NO difference between the two groups The alternative hypothesis (Ha) would be there is a difference

Examples of H0 ldquoproportion of survivors in population of patients treated by X= =proportion of survivors in population of patients treated by Yrdquo

ldquoH0 πX= πYrdquo

ldquomean weight in population of patients in your clinic=mean weight in

general populationrdquo ldquoH0 μ=μ0rdquo

10

The p-value Example H0 The coin is fair (ie will produce as many heads as tails) =gt population proportion π=05

Outcome 42 heads out of 100 times tossing =gt sample proportion p=042 Is the difference between 05 and 042 statistically significant or due to chance How do we decide what proportion of heads we might expect to get if the coin is fair Matter of interest P(heads le 42 | coin is fair)=0067 gt 005

=gt the coin is most likely fair P(heads le 41 | coin is fair) = 0044 lt 005 =gt the coin is most likely not fair p-value

p-value

if outcome 41 heads

11

Definition The p-value is defined as the probability of obtaining the result or a more extreme result if the null hypothesis is true Interpretation of p-value

We usually say if p le 005 then the results are

statistically significant (or unlikely due to chance) and there is strong enough evidence against the null hypothesis

12

Example

lt005

For Ceftazidim the difference in Odds Ratios for Denmark (OR=153) and Israel (OR=34) is statistically significant (plt005) For Ceftriaxone there is no evidence to conclude that the difference (between OR=80 and OR=37) is statistically significant (pgt005)

gt005

13th ICID Abstract 66004

Odds Ratios of antimicrobial resistance in hospital-acquired versus community-acquired infections

13

Type I and Type II Errors

Type I error (false positive) Rejecting H0 when it is true α (alpha) the probability of Type I error Significance level (the probability of Type I error that we are willing to accept) usually α = 5 Confidence level 1- α usually 95 Type II error (false negative) Accepting H0 when it is false β the probability of Type II error

14

Power

The probability of finding a specified difference or larger when a true difference exists (the probability of correctly rejecting the null hypothesis) Power = [1 - (probability of a Type II Error)] =1- β

15

To define Power

bull Power depends on 1048707 significance level (α) 1048707 sample size 1048707 minimum size of the effect that would be clinically useful bull Fix α (= 5) and then try to minimize β (maximize 1 - β) bull Keeping other variables constant the best way to increase power is to increase sample size (Studies with small sample sizes tend to have low power) bull Typical power is 80 or greater

bull You can compute the power for a study that you are about to run (good idea) or you can compute post-hoc power after an experiment is done

16

RCT of homeopathic arnica for post fracture healing

Example

Conclusion

17

gt 005 (= α) probability of observing 5 difference or more when the null hypothesis is true pgt5 =gt do not reject null hypotesis

18

=gt do more research with a larger sample

19

Test statistic

bull A function of the sample data on which the decision is to be based

bull Follows the general format

0Observed Value - Expected Value (H )

Standard Error

20

0

xt

s n

Test statistic Example

variability

difference in means

one sample t-test statistic

statistical difference

difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n

Sampling distribution of t under H0

The rejection region is the set of all values of the test statistic that

cause us to reject the null hypothesis

The t-distribution contains all possible values of t computed for random sample means selected from the population described by H0


critical value

H0 μ= μ0

t t

22

One(two)-tailed statistical test

One-tailed Ha μgtμ0

Two-tailed Ha μneμ0

α=05

α=025 α=025

23

The hypothesis testing process bull Use your research question to define H0 and Ha

bull Collect samples data and determine sample statistics (eg sample mean

sample proportionhellip) bull Choose the appropriate statistical test

bull Check assumptions for this test

bull Express H0 and Ha in mathematical terms

bull Decide on a level of significance (usually 5) bull Calculate test statistic

bull Refer test statistic to known distribution it would follow with H0

bull Calculate p-value (a probability of a test statistic arising as or more extreme than observed if H0=true)

bull If p-value lt significance level then reject H0 otherwise do not reject H0

24

There are many different statistical tests The choice of which test to use depends on several factors The type of data (continuous nominal etc) The distribution of the data (normally distributed or not) The type of sample statistics (mean median proportion ratio etc)

Statistical test

25

One Sample t test Purpose to compare the mean score of a sample to a known value Assumptions bull Random sample bull Metric sample data bull Normally distributed sample data (although this assumption is less critical when the sample size ge 30)

0

xt

s n

0observed value - expected value (H )test statistic=

standard error

H0 μ= μ0

One sample t test - example

bull Research question Are patients at your clinic have higher level of blood creatinine than the national average bull H0 creatinine level in population of men in your clinic = creatinine level in general male population bull For healthy men creatinine levels are normally distributed

the average creatinine level in male population is μ0=09 mgdL

bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28

bull Test one-sample t-test

bull a=5 bull H0 μ=09 Ha μne09

bull 2-tailed test bull Test statistics t = (12-09) 0088 = 34 (=gt the results are 34 standard error units higher the expected value)

bull Use a statistic table to look up the t-distribution and find critical value t

29

df

α

t distribution depends on degrees of freedom df = n-1

t=213 (depends on df=n-1=15)

30

t=34

P-value is the area in the tails greater than |t|=34

bull Calculate p-value

t=-34 Here p lt a=005

bull P-value lt a =gt reject the null hypothesis

We conclude that the result we saw was unlikely to have occurred by chance alone and that difference in mean creatinine levels in malersquos from the clinic and in a general male population is statistically significant (005 significance level)

t

31

One sample t test ndash in SPSS

plt5 =gt Reject Ho Difference is statistically significant

32



Alternatively Estimation

Mean is

unknown

Population Random Sample I am 95

confident that is between

101 amp 138

Mean

X = 12

Sample

CI = an interval of values that contains the true value of the parameter with specified confidence

34

95 CI for a population mean

(If is known)

bull Assumptions

ndash Population standard deviation () is known (this is hardly

ever true)

ndash Population is normally distributed or sample is large

n

σ961Xμ 95 Confidence Interval Estimate

35


( If is unknown)

bull Assumptions

ndash Population standard deviation ( ) is unknown =gt use

sample standard deviation (s)


95 Confidence Interval Estimate n

stXμ

Use a statistic table to look up the t-distribution and find critical value t

36

CI of the Mean in SPSS (rsquocreatininersquo example)

The 95 confidence interval does not include the comparison value (09) This is equivalent to rejecting the null hypothesis

37

Graphical presentation of a CI

38

CI of the Mean Difference does not include 0 lt=gt p-valuelt005 =gt reject H0

CI for a Mean Difference

39

In Denmark the CI of the OR contains 1 (08-18)

=gt despite a sample OR of nearly 4 there is no evidence to conclude

that hospital acquisition of a coagulase-negative staphylococci in Denmark is a statistically significant rsquoriskrsquo for resistance to penicillin

In Israel the CI of the OR does not contain 1 (13-38) =gt hospital acquisition of a coagulase-negative staphylococci in Israel is a statistically significant rsquoriskrsquo for resistance to penicillin


Odds Ratios of antimicrobial resistance in nosocomial versus non-nosocomial infections

CI for an Odds Ratio

40

CI of the OR in SPSS (rsquoApgar scorersquo example)

41



42

2

Learning material

Chapters 9-11 Confidence Interval Estimation Chapters 12-14 Testing hypotheses

3



4

Ideas and evidence


Case reports

Biology


observation

Imagination

Reasoning



Tests of hypothesis


IDEAS

5









6

Learning Objectives




7


Example




8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

3



4

Ideas and evidence


Case reports

Biology


observation

Imagination

Reasoning



Tests of hypothesis


IDEAS

5









6

Learning Objectives




7


Example




8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

4

Ideas and evidence


Case reports

Biology


observation

Imagination

Reasoning



Tests of hypothesis


IDEAS

5









6

Learning Objectives




7


Example




8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

5









6

Learning Objectives




7


Example




8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

6

Learning Objectives




7


Example




8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

7


Example




8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

8



9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

9






10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

10




p-value

if outcome 41 heads

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

11




12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

12

Example

lt005


gt005



13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

13



14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

14

Power


15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

15

To define Power



16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

16


Example

Conclusion

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

17


18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

18


19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

19

Test statistic




Standard Error

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

20

0

xt

s n


variability

difference in means



difference in means

variability

micro0

H0 μ= μ0

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

Rejection area

0

xt

s n






critical value

H0 μ= μ0

t t

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

22




α=05

α=025 α=025

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

23










24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

24


Statistical test

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

25


0

xt

s n


standard error

H0 μ= μ0




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42




bull Sample

_

x 12

s = 0352

se= 0088

mg dL

mg dL

mg dL

checking normality

metric data

random sample

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

28





29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

29

df

α



30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

30

t=34






t

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

31



32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

32




Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42


Mean is

unknown



101 amp 138

Mean

X = 12

Sample


34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

34


(If is known)

bull Assumptions


ever true)


n


35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

35


( If is unknown)

bull Assumptions





stXμ


36



37


38



39








40


41



42

36



37


38



39








40


41



42

37


38



39








40


41



42

38



39








40


41



42

39








40


41



42

40


41



42

41



42

MedI 7 Hypothesis Testing

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Transcript of MedI 7 Hypothesis Testing