Hypotheses Testing

Hypotheses Testing

Chapter 8

Hypothesis TestingFirst we state a hypothesis about a population. We hypothesize

that the mean IQ for registered voters in the US is μ = 110

Though we do not know it to be true, this is our hypothesis and we expect a sample from this population to have a mean ‘around’ 110. (around - because there could be error).

Next we select a random sample of n=200 and compute Mean IQ for the sample.

Finally we compare this Mean IQ to our prediction in the hypothesis. If the sample mean is consistent with the prediction, we will conclude that our hypothesis is reasonable, if there is a large discrepancy we will conclude that the Hypothesis is wrong.

SO,

Sample IQ = 135; Hypothesized IQ = 110 : Right or wrong?Sample IQ: 112; Hypothesized IQ = 110 : Right or wrong?

Hypothesis Testing

Independent variableVariable manipulated by the researcher

Dependant variableVariable that is observed in order to assess the effect of the treatment.

Control conditionReceive no experimental manipulation

Experimental conditionReceive experimental manipulation

Steps in Hypothesis TestingSteps in Hypothesis TestingStep1 - State the HypothesisStep1 - State the Hypothesis

• AA Statistical hypothesisStatistical hypothesis is a guess about a population parameter. This guess may or may not be true.

• The null hypothesisnull hypothesis, symbolized by H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value or that there is no difference between two parameters.

• The alternative hypothesisalternative hypothesis, symbolized by H1, is a statistical hypothesis that states a specific difference between a parameter and a specific value or states that there is a difference between two parameters.

Null Hypothesis: H0

• The null hypothesis is a statement about a value of a population parameter

• Must contain the condition of equality ( = )

• We test the null hypothesis directly

• The conclusions of the test are either “reject H0” or

“fail to reject H0”


Alternative Hypothesis: H1

• Must be true if H0 is false

• Contains the conditions of inequality

(i.e.: , < , or > )

• It is the opposite of the null hypothesis H0


Steps in Hypothesis Testing: Step-1Steps in Hypothesis Testing: Step-1 Example

A medical researcher is interested in finding out whether a new medication will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication, i.e. will there be changes in the pulse rate if someone takes the medicine.

Steps in Hypothesis Testing: Step-1Steps in Hypothesis Testing: Step-1 Example

Hypotheses: Will the pulse rate be different from th mean pulse rate of the population (82 beats per minute)

Null H0: = 82 , i.e. The mean pulse rate of the patients who take the medication is 82.

Alternate H1: 82, i.e. The mean pulse rate of the patients who take the medication is not 82

Steps in Hypothesis Testing: Steps in Hypothesis Testing: Step-1Example

A chemist invents an additive to increase the life of an automobile battery. If the mean lifetime of the battery is 36 months, then his hypotheses are:

– Null: H0: 36 In words: The mean lifetime of the battery is 36 months.

– Alternate: H1: 36 In words: The mean lifetime of the battery is greater than 36 months.

– This is a one-tailedne-tailed test because we are interested in mean lifetime of 36. We are only interested in the increase in lifetime of the battery.

Steps in Hypothesis Testing: Step-1Steps in Hypothesis Testing: Step-1 A contractor wishes to lower heating bills by using a special type of insulation in houses. If the average of the monthly heating bills is $78, her hypotheses about heating costs will be

Null: H0: $78 In words: The mean heating bill will be equal to or more than &78.

Alternate: H1: $78 In words: The mean heating bill is less than $78.

This is a one-tailed-tailed test because we are interested in heating bills <$78 We are only interested in the decrease in heating bills.

Steps in Hypothesis TestingStep-2: Set the criteria for a decision

We set the criteria or region for acceptance of H0. This

criteria or level is known as the alpha level (α) or level of significance. In research

we commonly find α = .05 (5%) or α = .01 (1%)

or α = .001 (0.1%)


Alpha level or the level of significance is the probability value that is used to define the very unlikely sample outcomes if the null hypothesis is true. Denoted by symbol α (Greek letter alpha).

Critical region is composed of extreme sample values that are very unlikely to be obtained if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level that we set for our research. If sample data fall in the critical region, the null hypothesis is rejected.


The critical region (very unlikely outcomes) for = .05.


The locations of the critical region boundaries for three different levels of significance: = .05, = .01, and = .001.

Steps in Hypothesis TestingStep-3: Collect Data and Compute Sample

Statistics

We have so far learnt one type of sample statistic:

z = (M – μ)/σM

Stated in Words:z = sample mean – hypothesized population mean

Standard error between Mean and μ

Steps in Hypothesis TestingStep-4: Make a decision

Use the z-score value obtained in previous step to make a decision about the null hypothesis.

Check if the obtained z-score falls within the critical region:

If yes – reject null hypothesis H0 and support alternate hypothesis H1

If no – fail to reject null hypothesis, so alternate hypothesis is false or unlikely.

Steps in Hypothesis TestingStep-4: Make a decision

n = 25; μ=80; M=92; σ = 20; σM = 4;

Z = (M- μ)/ σM = (92-80)/4 = 12/4 = 3.00

Now, if α = .05; what is the critical value of the z-score?

Where does our z-score fall?

So – reject or accept the null?

Hypotheses Testing

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