Standard error of proportion and standard error of difference between two proportions

Standard Error of Proportion

Standard error is a measure of chance variation.
Standard error of proportion denoted by SEp
It is calculated by the formula

Uses of Standard Error

From the Standard error of proportion, 95% confidence limit can be calculated as follows:-

95% Confidence Limits =proportion±2 standard error

95% Confidence Limits =p±2 SEp

Standard error of proportion is used in Z test to check if there is a significant difference between sample proportion and population proportion or between two samples. In case of comparison with the population the standard error of population is calculated using the population prevalence and sample size of study sample. In case of comparison between two samples, the standard error of difference between two proportions is to be calculated.

Standard error of proportion can also be used for calculating sample size.

Example 1:

The non-communicable disease survey in Pratikshanagar found the prevalence of hypertension as 12% in the study conducted in 400 adult population, Calculate the 95% confidence limits.

Standard Error of Difference between two Proportions

Example 2:

The non-communicable disease survey in Pratikshanagar found the prevalence of hypertension as 12% in the study conducted in 400 adult population. The study in Vidyavihar showed the prevalence of hypertension as 6% in 200 adult individuals. Calculate the standard error of difference between two proportions.

Tests of Significance for Qualitative Data

To test if the difference observed between two sample proportions or between sample proportion and population proportion is significant, Z test can be applied.

When you want to check if two variables have an association then, chi-square test is performed.

Z test

Interpretation of Z calculated value

Z cal >1.96 at 5% level of significance, reject null hypothesis

Z cal <1.96 at 5% level of significance, accept null hypothesis

Usually, a two tailed hypothesis and 5% level of significance is considered for which the Z table value is 1.96. If you consider 1% level of significance in a two tailed hypothesis, the Z table value is 2.58.

In case of one tailed hypothesis, and a 5% level of significance the Z table value is 1.64. If you consider 1% level of significance in a one tailed hypothesis, the Z table value is 2.33.

Example 3:

The non-communicable disease survey in Pratikshanagar found the prevalence of hypertension as 12% in the study conducted in 400 adult population. Prevalence of hypertension in the district is 10%. Does the prevalence of hypertension in Pratikshanagar vary significantly from the district population.

Step 1: State Null Hypothesis – H0 and State Alternative Hypothesis – H1

H0 = There is no significant difference in the prevalence of hypertension

between Pratikshanagar and District population.

H1 = There is significant difference in the prevalence of hypertension between Pratikshanagar and district population.

Step 2: State the level of significance

Level of significance is 5%

Step 3: Choose the appropriate Test of Significance (ToS)

Appropriate test of significance is Z test.

Justification for choosing Z test

Type of data – Qualitative

Sample Size >30

No. of groups – one

Data assumed to be normally distributed

Step 4: Calculate the test statistic (calculated value)

Now apply Z test

Sample proportion = p = 12%

Population proportion = P=10%

Standard error of proportion = 1.5

Step 5: Compare the test statistic i.e. calculated Z or t value with the

table value to draw inference.

Table value of Z at 5% level of significance is 1.96

As Z calculated value (1.33) < Z table value (1.96), accept null hypothesis.

There is no significant difference between the prevalence of hypertension in Pratikshanagar and district.

Example 4:

Step 1: State Null Hypothesis – H0 and State Alternative Hypothesis – H1

H0 = There is no significant difference in the prevalence of hypertension

between Pratikshanagar and Vidyavihar.

H1 = There is significant difference in the prevalence of

hypertension between Pratikshanagar and Vidyavihar.

Step 2: State the level of significance

Level of significance is 5%

Step 3: Choose the appropriate Test of Significance (ToS)

Appropriate test of significance is Z test.

Justification for choosing Z test

Type of data – Qualitative

Sample Size >30

No. of groups – two

Data assumed to be normally distributed

Step 4: Calculate the test statistic (calculated value)

Step 5: Compare the test statistic i.e. calculated Z value with the table value to draw inference.

Table value of Z at 5% level of significance is 1.96

As Z calculated value (2.55) > Z table value (1.96), accept null hypothesis.

There is significant difference between the prevalence of hypertension in Pratikshanagar and Vidyavihar.

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