Z test

DR IRFAN MOMIN

Z test

Dr. Muhammedirfan H. MominAssistant ProfessorCommunity Medicine DepartmentGovernment Medical College, Surat.

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INTRODUCTION

The use of any statistical method for analyzing data is based on the objectives of the study, the hypothesis to be tested and type of available statistical data which is to be analyzed to answer the research questions

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Statistical TestsHow to tell if something (or somethings) is

different from something else

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TYPES OF DATA

It can be - Primary or Secondary- Qualitative or Quantitative

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• Primary data is that which is collected by the researcher to address the current research.

• Secondary data refers to data gathered by others or from other studies like official records, publications, documents etc.

• Qualitative data refers to data having counting of the individuals for the same characteristic and not by measurement.

• Quantitative data refers to data having magnitude and the characteristic is measured either on an interval or on a ratio scale.

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Qualitative ( Sex, Religion)• Data types Quantitative

Continuous Discrete (measurable) (countable)

Age No. of. Children Hb No. of Cases

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Pt. No. Hb. Pt. No. Hb. Pt. No. Hb. 1 12.0 11 11.2 21 14.9 2 11.9 12 13.6 22 12.2 3 11.5 13 10.8 23 12.2 4 14.2 14 12.3 24 11.4 5 12.3 15 12.3 25 10.7 6 13.0 16 15.7 26 12.7 7 10.5 17 12.6 27 11.8 8 12.8 18 9.1 28 15.1 9 13.5 19 12.9 29 13.4 10 11.2 20 14.6 30 13.1

SAMPLE DATA SET

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Hb (g/dl) No. of patients

9.0 – 9.9 110.0 – 10.9 311.0 – 11.9 612.0 – 12.9 1013.0 – 13.9 514.0 – 14.9 315.0 – 15.9 2

Total 30

TABLE FREQUENCY DISTRIBUTION OF30 ADULT MALE PATIENTS BY Hb

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Table 1 Risk factors for Myocardial Infarction for patients (n=57) admitted to the Kilpauk Medical College Hospital, Chennai, Jan- Sep 1998

MI Patients Risk factor No %

Hypertension Smoking Diabetes CAD Hyperlipedemia None

24 20 13 9 2 8

42.1 35.1 22.8 15.8 3.5

14.0

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Steps in Test of Hypothesis1. Determine the appropriate test 2. Establish the level of significance3. Formulate the statistical hypothesis4. Calculate the test statistic5. Compare computed test statistic against

a tabled/critical value

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1. Determine Appropriate Test

• Z Test for the Mean • (Standard error of difference between two

means)• Z Test for the Proportion• (Standard error of difference between two

proportions)

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2. Establish Level of Significancep is a predetermined valueThe convention

p = .05 p = .01 p = .001

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NORMAL DISTRIBUTION

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2-tailed regions

A sampling distribution for H0 showing the region of rejection for a = .05 in

a 2-tailed z-test.

Fig 10.4 (Heiman p. 228)

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1-tailed region, above mean


a 1-tailed z-test.


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1-tailed region, below mean

A sampling distribution for H0 showing the region of rejection for a = .05 in a 1-tailed z-test where a decrease in

the mean is predicted.


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If P < 0.05, the observed difference is‘SIGNIFICANT (Statistically)’

P< 0.01, sometimes termed as ‘Highly Significant’

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INTERPRETATION OF SIGNIFICANCE

SIGNIFICANT Does not necessarily mean that the

observed difference is REAL or

IMPORTANT. Only that it is unlikely

(< 5%) to be due to chance.

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INTERPRETATION OF NON - SIGNIFICANCE

NON - SIGNIFICANT Does not necessarily mean that there is no real difference; it means only that the observed difference could easily be due to chance

(Probability of at least 5%)

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3. Determine The Hypothesis:Whether There is an Association or Not

Write down the NULL HYPOTHESIS and ALTERNATIVE HYPOTHESIS and set the LEVEL OF SIGNIFICANCE.

Ho : The two variables are independentHa : The two variables are associatedWe will set the level of significance at

0.05.

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For ExampleSome null hypotheses may be:

‘there is no relationship between the height of the land and the vegetation cover’.

‘there is no difference in the location of superstores and small grocers shops’

‘there is no connection between the size of farm and the type of farm’

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TEST FOR PROPORTIONS

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STANDARD ERROR OF DIFFERENCE BETWEEN TWO PROPORTIONS

P1 - P2

Z = ------------------

SE (P1 - P2)

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SE (P1 - P2) = P1Q1 + P2Q2

n1 n2

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EXAMPLEIf swine flu mortality in one sample of 100 is 20%

and in another sample of 100 it is 30%. Is the difference in mortality rate is significant?

P1= 20 q1 = 80 n1 = 100P2= 30 q2 = 70 n2 = 100

P1 - P2

Z = ------------------ SE (P1 - P2)

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SE (P1 - P2) = 20x80 + 30x70

100 100

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SE (P1 - P2) = 37 = 6.08

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P1= 20 q1 = 80 n1 = 100P2= 30 q2 = 70 n2 = 100

P1 - P2

Z = ------------------ SE (P1 - P2)

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P1= 20 q1 = 80 n1 = 100P2= 30 q2 = 70 n2 = 100

20-30 Z = --------- = -1.64 6.08

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Z value and probability

z 1.96 2.58p 0.05 0.01

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Obtained z value (1.64) is less than critical z value (1.96) , so P >0.05, hence difference is insignificant at 95 % confidence limits.

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STANDARD ERROR OF DIFFERENCE BETWEEN TWO MEANS

X1 - X2 Z = ------------------

SE (X1 - X2)

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SE (X1 - X2) = SD12 + SD22

n1 n2

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EXAMPLEIn a nutritional study, 100 children were given usual diet and 100

were given vitamin A and D tablets. After 6 months, average weight of group A was 29kg with SD of 1.8kg and average weight of group B was 30kg with SD of 2kg. Is the difference is significant?

SD1= 1.8 n1 = 100SD2= 2 n2 = 100

X1 - X2

Z = ------------------ SE (X1 - X2)

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SE (X1 - X2) = SD12 + SD22

n1 n2

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SE (X1 - X2) = (1.8)2 + (2)2

100 100

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SE (X1 - X2) = 3.24+4 100

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SE (X1 - X2) = 0.0724 =0.27

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SD1= 1.8 n1 = 100SD2= 2 n2 = 100

X1 - X2

Z = ------------------ SE (X1 - X2)

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SD1= 1.8 n1 = 100SD2= 2 n2 = 100

29 - 30 Z = ---------------- = -3.7 0.27

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2-tailed regions


a 2-tailed z-test.


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Z value and probability

z 1.96 2.58p 0.05 0.01

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As obtained value of z (-3.7) is higher than critical value (-1.96 or -2.58), the observed difference is highly significant, vitamins played a role in weight gain .

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