Hypothesis testing

Draw inferences about a population based on a sample

Null Hypothesis expresses no difference

Example:

H0: = 0Often said “H naught” Or any number

Later…….H0: 1 = 2

Alternative Hypothesis

H0: = 0; Null Hypothesis

HA: = 0; Alternative Hypothesis

Researcher’s predictions should be a priori, i.e. before looking at the data

-To test a hypothesis about , determine X from a random sample

-If H0 is true, what is the probability of X as far (above or below- 2 tailed) from as the observed X (for a given n)?

-Calculate the normal deviate

Z = X - x

Normal deviate

Population Mean

SE of mean

Sample Mean

How to determine what proportion of a normal population lies above/below a certain level

120 cm

The average Hobbit

If distribution of Hobbit heights is normal with mean = 120 cm, SD = 20

Half < 120 & half >120

What is probability of finding a Hobbit taller than 130 cm??

Calculate the normal deviate

Z = Xi -

- Normal deviate- Test statistic

- Any point on normal curve- Here, 130 cm Mean

SD

Z = (130-120)/20 = 0.5

- Calculate P (Z); table 2.B Zar, Table A in S&R)

-If Z is large, the probability that H0 is true is small

-Pre-selected probability level, , that you require to reject the null, referred to as significance level

-0.05 is common

-If Z (test statistic) is larger than critical value, then H0 is rejected

-If Z (test statistic) is smaller than critical value, then H0 is not rejected (failure to reject null)

P (probability) (Xi >130 cm) = P (Z>0.50) = 0.3085 or 30.85%

What is the probability of finding a hobbit between 120cm and 130cm tall?

Table B.2; ZarTable A S & R

de

ns

ity

120cm 130cm

Area = 0.3085 or 30.85%

0 0.5

de

ns

ity Area = 0.3085

or 30.85%

BE AWARE!!

Different tables will give you different areas under the curve. You need to know what the table you are looking at is actually telling you!!

S&R Table A Your book gives you this area (0.1915)

You want this area: 0.5 - 0.1915=0.3085

120cm 130cm

Statistical Error

Sometimes H0 will be rejected (based on large test statistic & small P value) even though H0 is really true

i.e., if you had been able to measure the entire population, not a sample, you would have found no difference between and some value- but based on Xbar you see a difference.

The mistake of rejecting a true H0 will happen with frequency

So, if H0 is true, it will be rejected ~5% of the time as frequently = 0.05

0

0 20

Population mean = 0

Sample mean = 20

Conclude based on sample mean that population mean 0, but it really does (H0 true), therefore you have falsely rejected H0

Type 1 Error

population=“True”

Sample=What you see

H0 : mean = 0

Statistical Error

Sometimes H0 will be accepted (based on small test statistic & large P value) even though H0 is really false

i.e., if you had been able to measure the entire population, not a sample, you would have found a difference between and some value- but based on Xbar you do not see a difference.

The mistake of accepting a false H0 will happen with frequency β

0

Sample mean = 00 20

Sample mean = 20

Conclude based on sample mean that population mean = 0, but it really does not (H0 really false), therefore you have falsely failed to reject H0

Type 2 Error

Population= “True”

Sample= what you see

H0 : mean = 0

20

Finicky Words

Reject the null hypothesis (or other)

Fail to reject the null hypothesis

Prove the null hypothesis to be true

Accept the null hypothesis

Support the null hypothesis

Statistically correct

I think these are OK

H0 is true H0 Is not true

H0 Is rejected Type I error No error

H0 Is not rejected No error Type II error

Probability of Type I Error=

Probability of Type II Error=

rarely known or reported

power of a test = (1- ) = probability of rejecting null hypothesis that is really false and that should be rejected

For a given N, and inversely related

Both types of Error go down as you increase N

Read pgs 157-169 in S&R

A few more words on hypothesis testing

Methods trace to R.A. Fisher and colleagues. Before this, opinion of expert was criterion.

Many practicing / publishing statisticians take issue with the null hypothesis and testing framework (see upcoming quotes)

Still the dominant paradigm of analysis and you have to learn it

“Under the usual teaching, the trusting student, to pass the course must forsake all the scientific sense that (s)he has accumulated so far, and learn the book, mistakes and all." (Deming 1975)

"Small wonder that students have trouble [with statistical hypothesis testing]. They may be trying to think." (Deming 1975)

"... surely, God loves the .06 nearly as much as the .05." (Rosnell and Rosenthal 1989)

"What is the probability of obtaining a dead person (D) given that the person was hanged (H); that is, in symbol form, what is p(D|H)? Obviously, it will be very high, perhaps .97 or higher. Now, let us reverse the question: What is the probability that a person has been hanged (H) given that the person is dead (D); that is, what is p(H|D)? This time the probability will undoubtedly be very low, perhaps .01 or lower. No one would be likely to make the mistake of substituting the first estimate (.97) for the second (.01); that is, to accept .97 as the probability that a person has been hanged given that the person is dead. Even thought this seems to be an unlikely mistake, it is exactly the kind of mistake that is made with the interpretation of statistical significance testing---by analogy, calculated estimates of p(H|D) are interpreted as if they were estimates of p(D|H), when they are clearly not the same." (Carver 1978)

One sample, two tailed tests concerning means

Does the body temperature of a group of 24 crabs differ from room temperature

TweetyBird parakeet food company wants to know if their mega-bird formulation helps birds grow. They measure the weight gain/loss of 40 birds eating the food for one week. Does this differ from zero.

Describe other scenarios…..

Null; H0: = room temp

Alternative; HA: ≠ room temp

One sample, two tailed tests concerning meansCrab Temperature

Must determine if sample mean (xbar) is different from room temp

Similar to calculating normal deviate, calculate “t”

t = X -

s x

t-statistic

Value to which you compare

Sample SE

Sample Mean

William Sealy Gosset (1876 –1937)

Mathematician worked as brewer for Guinness

Guinness progressive agro-chemical business, Gosset applied statisticas in brewery and farm selection of best varieties of barley.

Guinness prohibited publishing by employees due to worry over trade secrets

Used pseudonym “Student” and his most famous achievement now referred to as the Student t-distribution

Gosset was a friend of both Pearson and Fisher, an achievement, for each had a massive ego and a loathing for the other. Gosset was a modest man who cut short an admirer with the comment that “Fisher would have discovered it all anyway.”

One sample, two tailed tests concerning meansCrab Temperature

For the crabs: = .05 (set by you ahead)=24.3 CXbar= 25.03 C = 24 (n-1)S2(variance)= 1.80 C2

s x=

sx =s

n

1.80 C2

25

sx =s2

n

t = X -

s x

Sample SE SD VarianceMean SS

t = X -

s x

t = 25.03 C – 24.3 C

0.27 C2

= 2.704

t0.05(2) = 2.064 (critical t from table B in S&R, B3 in Zar)

test statistic (t) > critical t …….

Reject null hypothesis, conclude sample did not come from population with body temp of 24.3

Excel demo

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t for = 24, =0.05

0 2.064-2.064

Critical value

t- distribution = normal distribution for very large sample sizes

Area outside critical values represent 5% total area

Expect xbar so far from that it lies in critical area only 5 % of time

2.704=t

2.5%2.5%

If p<0.05 (or your chosen ), expect to get a values as extreme as the observed based on chance alone 5% (or your chosen %) of the time

Expect to falsely reject null ~5% of the time

H0 is true H0 Is not true

H0 Is rejected Type I error No error

H0 Is not rejected

No error Type II error

Theoretical basis of t testing assumes that the sample came from a normal population

But….. minor deviation from normality not does not affect validity, ie test is “robust from deviation from normality”

Effect of deviation from normality more important with small

Effect of deviation from normality decreases as N increases

Assumptions of a t-test

Assumes that data are random sample

Data must be true replicates (can’t measure the same crab 25 times; Hurlbert 1984, more later)

One sample, one tailed tests concerning means

Does a drug cause weight loss?

The Jamesville county school board has mandated that the mean standardized reading test scores should be above 440. Oak elementary school wants to know if their mean test score > 440.

Describe other scenarios…..

One sample, one tailed tests concerning meansWeight loss product

Null; H0: ≥ 0; weight gain or no change, ie no loss

Alternative; HA: <0; weight loss

Must determine if sample mean weight gain (xbar) is different from 0

One sample, one tailed tests concerning meansWeight loss

For weight loss: = .05 (set by you ahead)=0 kgXbar= -0.61 kg = 11 (n-1)S2(variance)= 0.4008kg2

s x=

sx =s

n

0.4008kg2

12

sx =s2

n

t = X -

s x

Sample SE SD VarianceMean SS

t = X -

s x

t = -.61kg -0kg

0.18 kg= -3.389

t0.05(11) = 1.796 (critical t from table B3; table gives you critical t for 2-tails)

test statistic (t) > critical t …….

Reject null hypothesis, support alternative hypothesis of weight loss

Excel demo

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t for = 24, =0.05

0 2.064-2.064

Xbar expected in tails only 5% of the time, then 95% of the time Xbar lies in this region

Confidence limits of mean

So, if we know xbar and SE and degrees of freedom, we can calculate an interval in which we will be 95% (or other value) confident that the “true mean” () falls

Confidence limits of mean

X ± t(2), * sx

For the crabs…..

CI= mean ± (critical t * SE)

25.03 ± (2.064 * .27)25.03 ± 0.56

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t for = 24, =0.05

25.03 25.5924.47

Xbar expected in tails only 5% of the time, then 95% of the time Xbar lies in this region

Confidence limits of mean

CI is two tailed

The smaller SE, the smaller CI. We have more precise estimate of when SE small

A large N results in smaller SE

Parameter estimate from large sample more precise than from small sample