Inference Concepts Hypothesis Testing. Confidence IntervalsSlide #2 Inference Sample Statistic...

Inference Concepts

Hypothesis Testing

Confidence Intervals Slide #2

Inference

SampleStatistic

PopulationParameter

• Hypothesis/Significance Testing– assess evidence the data gives against a hypothesis

about a parameter

• Confidence Regions– provide a range (region) believed to contain the

parameter with a certain believability (confidence)

Inference Concepts Slide #3

The Scientific Method

Make Observation

Make Predictions from Hypothesis

Gather Observations / Experiment

Compare Observations to PredictionsMatch, gain belief in hypothesis

Don’t match, lose belief in hypothesis

Construct Hypothesis


The Scientific Method

Make Observation

Make Predictions from Hypothesis

Gather Observations / Experiment

Compare Observations to PredictionsMatch, gain belief in hypothesis

Don’t match, lose belief in hypothesis

Construct HypothesisStatistical hypothesis testing is at centerCompares predictions to observations in the face of sampling variability

Statistical hypothesis testing follows same logicCompares observations to predictions from a hypothesis


Two Main Hypothesis Types• Research Hypothesis

– a general statement of an effect

• Statistical Hypothesis (two types)– Alternative Hypotheses (HA)

• a mathematical representation of the research hypothesis• one of HA: parameter <,>, specific value

– Null Hypotheses (Ho)• the “no effect” or “no difference” situation• always Ho: parameter = specific value

Inference Concepts

What is the null and alternative hypotheses for these research hypotheses …

• “The mean density of Canada yew (Taxus canadensis) in areas not exposed to moose (Alces alces) on Isle Royale will be more than 1 stem per m2”

• “The mean age of medical college students (Homo sapien) is less than 24 years”

• “The mean number of murders per burrough is less than 90”

• “The mean longevity of employees at the company is different than 10 years”

Slide #6


An Example

• A research paper claims that the mean fetal heart rate is 137 bpm. A doctor feels that the mean rate is lower for women admitted to her clinic.

What are the statistical hypotheses?

HA: m < 137 HO: m = 137

• She will test her belief with … – a random sample of 100 patients– assuming s =10


The Null Hypothesis

• Assumed, initially, to be true– Used to predict what will be observed in a sample

• Thus, if H0: m=137 then one predicts that`x=137

• IF there was no sampling variability, then what do you think about H0 if`x=135 is observed


Does 135 support H0?

Does 134 support H0?

Does 136.5 support H0?

137138 139 140136135134

Sampling Variability

Assume H0 is true

100

10137,N~x

100

10137

SE – measure of sampling variability

x


Objectivity – p-value

• PR(observed statistic or value more extreme assuming H0 is true)

– shade to left if a “less than” HA

– shade to right if a “greater than” HA

– shade into both tails if a “not equals” HA

x

x

x

On

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An Example

Suppose`x=135.9 was observed

p-value = Pr(`x=135.9 or less, if m=137 )

137 138 139 140136135134135.9

p-value = 0.1357

distrib(135.9,mean=137,sd=1)

100

10137,N~x

100

10137

HA: m < 137


Objectivity – p-value

• Compare to rejection criterion – a– if p-value < a then reject H0

– if p-value > a then Do Not Reject (DNR) H0

• Rejection criterion (a)– “sets” cut-off value for determining support of H0

– Set by researcher a priori– typical values are 0.10, 0.05, 0.01

• PR(observed statistic or value more extreme assuming H0 is true)

Critical


An Example

• The doctor set a at 0.05

• The p-value of 0.1357 is greater than a– Thus, DNR H0

– Conclude that mean fetal heart rate for all of her patients is not less than 137 bpm• The`x of 135.9 likely occurred because of sampling

variability and not a real difference from 137 bpm

137135.9x

Inference Concepts

For each situation below, write a definition of the p-value, compute the p-value, and make a decision

– HA: m<80, s=40, n=50, `x=74, a=0.05

– HA: m>100, s=20, n=40, `x=105, a=0.05

– HA: m>100, s=20, n=80, `x=105, a=0.05

– HA: m≠100, s=20, n=60, `x=103, a=0.10

– HA: m<100, s=20, n=40, `x=96, a=0.01

Slide #14


Summary• Make statistical hypotheses from research hypothesis

• Use H0 to make prediction (Assume H0 is true)– this is why H0 must be the “equals” situation

• Compare predicted statistic to observed statistic– calculate p-value

• Compare p-value to rejection criterion (a)– if p-value > a then DNR H0

• conclude that H0 could be correct

– if p-value < a then reject H0

• conclude that H0 is probably not correct

Critical


if p-value > a then H0 could be correct• Recall that `x = 135.9

• for H0: m=137, p-value=0.1357, DNR H0

• for H0: m=137.1, p-value=0.1151, DNR H0



• There are always several other hypotheses that would also not be rejected.

distrib(135.9,mean=137,sd=1)

distrib(135.9,mean=137.1,sd=1)




DNR vs Accept

• The data do not contradict this H0, but it is not fully known if this H0 is true.


137 138 139 140136135134

if p-value < a then H0 is probably incorrect

Even if H0 is truly correct it is possible to observe a statistic in the tail, resulting in a p-value < a, and a rejection of H0.

x


Truth about Popn

Ho True Ho False

RejectHo

Decisionfrom

Sample DNR HoType II

b

Type Ia

Correctpower

Correct

Decision Making Errors

Set a priori by the researcher

Can’t be known, because truth is not known


Effects on b• b is inversely related to a

– i.e., “trading errors”

• b is inversely related to n– i.e., “more information means fewer errors”

• b is inversely related to difference between true and hypothesized value of parameter– i.e., “more obvious difference means fewer errors”

Inference Concepts Hypothesis Testing. Confidence IntervalsSlide #2 Inference Sample Statistic...

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