Hpothesis Tests NET

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HYPOTHESIS TESTS &

CONTROL CHARTS

By S.G.M.







Testing Hypothesis

The assumption wish to test is called Null Hypothesis---Ho

If our sample result fail to support the Null hypothesis ,we must

conclude that something else is true

Whenever we reject the hypothesis the conclusion we do accept iscalled the alternative hypothesis and is symbolised as H1.

ex; Ho:µ=200 ³ The null hypothesis is that population mean is equal to 200

We will consider the three possible alternative hypothesis:

H1:µ 200= The alternative hypothesis is that population mean is not equal to 200

H1:µ > 200= The alternative hypothesis is that population mean is greater than 200

H1:µ < 200= The alternative hypothesis is that population mean is less than 200



Null Hypothesis

(H0) is true

He truly is not guilty

Alternative Hypothesis

(H1) is true

He truly is guilty

Accept Null Hypothesis Right decision Wrong decisionType II Error

Reject Null Hypothesis Wrong decision

Type I Error

Right decision

Purpose of Hypothesis testing is not to question the computed value of the sample

statistic but to make a judgement about the difference between that samplestatistic and hypothesized population parameter.



Interpreting the significance level

the significance level is the criterion used for rejecting the null hypothesis

The significance level of a statistical hypothesis test is a fixed probability

of wrongly rejecting the null hypothesis H0, if it is in fact true.

It is the probability of a type I error and is set by the investigator inrelation to the consequences of such an error. That is, we want to make

the significance level as small as possible in order to protect the null

hypothesis and to prevent, as far as possible, the investigator from

inadvertently making false claims.

The significance level is usually denoted by

Significance Level = P(type I error) =

Usually, the significance level is chosen to be 0.05 (or equivalently, 5%).



Power of a statistical hypothesis

The power of a statistical hypothesis test measures thetest's ability to reject the null hypothesis when it isactually false - that is, to make a correct decision.

In other words, the power of a hypothesis test is theprobability of not committing a type II error. It iscalculated by subtracting the probability of a type IIerror from 1, usually expressed as:

The maximum power a test can have is 1, the minimum is0. Ideally we want a test to have high power, close to 1.

Power = 1 - P(type II error) =



One-sided Test

A one-sided test is a statistical hypothesis test in which the values forwhich we can reject the null hypothesis, H0 are located entirely inone tail of the probability distribution.

In other words, the critical region for a one-sided test is the set ofvalues less than the critical value of the test, or the set of valuesgreater than the critical value of the test.

A one-sided test is also referred to as a one-tailed test ofsignificance.

The choice between a one-sided and a two-sided test is determinedby the purpose of the investigation or prior reasons for using a one-sided test.



Example

Suppose we wanted to test a manufacturers claim that there are, on average,50 matches in a box. We could set up the following hypothesis

H0: µ = 50,

against

H1: µ < 50 or H1: µ > 50

Either of these two alternative hypotheses would lead to a one-sided test.Presumably, we would want to test the null hypothesis against the first

alternative hypothesis since it would be useful to know if there is likely to beless than 50 matches, on average, in a box (no one would complain if theyget the correct number of matches in a box or more).

Yet another alternative hypothesis could be tested against the same null,leading this time to a two-sided test:

H0: µ = 50,

against

H1: µ not equal to 50

Here, nothing specific can be said about the average number of matches in abox; only that, if we could reject the null hypothesis in our test, we wouldknow that the average number of matches in a box is likely to be less than orgreater than 50.





Example

Suppose we wanted to test a manufacturers claim that there are, on average,50 matches in a box. We could set up the following hypothesis

H0: µ = 50,

against

H1: µ < 50 or H1: µ > 50

Either of these two alternative hypothesis would lead to a one-sided test.

Presumably, we would want to test the null hypothesis against the firstalternative hypothesis since it would be useful to know if there is likely to beless than 50 matches, on average, in a box (no one would complain if theyget the correct number of matches in a box or more).

Yet another alternative hypothesis could be tested against the same null,leading this time to a two-sided test:

H0: µ = 50,

against

H1: µ not equal to 50

Here, nothing specific can be said about the average number of matches in abox; only that, if we could reject the null hypothesis in our test, we wouldknow that the average number of matches in a box is likely to be less than orgreater than 50.



Stats: Type of Tests

how to determine if the test is a left tail, right tail, or two-tail test.

Left Tailed Test

H1: parameter < valueNotice the inequality points to the left

Decision Rule: Reject H0 if t.s. < c.v.

Right Tailed Test

H1: parameter > value

Notice the inequality points to the right

Decision Rule: Reject H0 if t.s. > c.v.

Two Tailed TestH1: parameter not equal valueAnother way to write not equal is < or >

Notice the inequality points to both sidesDecision Rule: Reject Ho if t.s. < c.v. (left) or t.s. > c.v. (right)

Reject Ho if the test statistic falls in the critical region



T-test-1908 by William Sealy Gosset

The t-test (or student's t-test) gives an indication of the separateness oftwo sets of measurements, and is thus used to check whether two sets ofmeasures are essentially different.

The t-test assumes:

A normal distribution (parametric data) Underlying variances are equal

It is used when there is random assignment and only two sets ofmeasurement to compare.

t = experimental effect OR t = difference between group means

variability standard error of difference between

group means



Z-test

The Z-test compares sample and population meansto determine if there is a significant difference.

It requires a simple random sample from a

population with a Normal distribution and wherethe mean is known.

Z= Sample mean-population mean

standard error of the mean The z value is then looked up in a z-table. A negative z value

means it is below the population mean (the sign is ignored inthe table).



F-Test---------Ronald A. Fisher

An F-test is any statistical test in which the test statistic

has an F-distribution under the null hypothesis.

It is most often used when comparing statistical models

that have been fit to a data set, in order to identifythe model that best fits the population from which the

data were sampled.

E

xactF

-tests mainly arise when the models have beenfit to the data using least squares.

OR



F-test

Since F is formed by chi-square, many of the chi-

square properties carry over to the F distribution.

The F-values are all non-negative

The distribution is non-symmetric

The mean is approximately 1

There are two independent degrees of freedom, one

for the numerator, and one for the denominator. There are many different F distributions, one for each

pair of degrees of freedom.



ANOVA-(t-test problems are solved)

Analysis Of Variance (ANOVA) overcomes these

problems by using a single test to detect significant

differences between the treatments as a whole.

(A significant problem with the t-test is that we typically accept significance

with each t-test of 95% (alpha=0.05). For multiple tests these accumulate and

hence reduce the validity of the results.)

ANOVA assumes parametric data.

Types of ANOVA 'One way' means one independent variable.

'Two way' means two independent variables.



Chi-square test

measures the alignment between two sets of frequency measures. These must

be categorical counts and Chi-square is reported in the following form:

c2 (3, N = 125) = 10.2, p = .012

Where:

3 - the degrees of freedom

125 - subjects in the sample

10.2 - the c2 test statistic

.012 - the probability of the null hypothesis being true ot percentages or ratios measures

Chi-squared, c2 = ( (observed - expected)2 / expected)

c2 = SUM( (fo - fe)2 / fe )

...where fo is the observed frequency and fe is the expected frequency.



Goodness of fit-0.05, 0.01 or 0.001

A common use is to assess whether ameasured/observed set of measures follows anexpected pattern.

The expected frequency may be determined from priorknowledge (such as a previous year's exam results) orby calculation of an average from the given data.

The null hypothesis, H0 is that the two sets of measuresare not significantly different.



Independence-0.95 or 0.99

The chi-square test can be used in the reverse manner togoodness of fit. If the two sets of measures are compared,then just as you can show they align, you can also determineif they do not align.

The null hypothesis here is that the two sets of measures aresimilar.

The main difference in goodness-of-fit vs. independenceassessments is in the use of the Chi Square table. Forgoodness of fit, attention is on 0.05, 0.01 or 0.001 figures.For independence, it is on 0.95 or 0.99 figures (this is whythe table has two ends to it).





Data that represents

the absence orpresence of

characteristics.

Data that contains a

range of quantities.Most measurements

yield variable data.

ATTRIBUTE DATA VARIABLE DATA



X-Chart

A centre line

UCL

LCL

+ 3sigma - control limits should contain most of the

observation



R-chart

The control chart that tracks sample ranges over

time.

An R chart is used with variable data.



P-chart

The control chart that tracks the percentage of

nonconforming items.

A P chart is used with attribute data.



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