Sample size FOR ONE SAMPLE

DR HAR ASHISH JINDALJR

PGIMS Rohtak

Sampling distribution

Estimating the population proportion

• Unknown proportion in the population is denoted by P

• The sampling distribution a of the sample proportion "p" is approximately normal with mean: E(p )= P, and variance: Var(p)=P(1-P)/n.

Sample size

• quantity d denotes the distance, in either direction, from the population proportion and may be expressed a

• z represents the number of standard errors away from the mean.

• quantity d is termed the precision(small as desired by simply increasing the sample size n.)

• Rearrange to get n

Sample size

• when the researcher has no idea as to what the level of P is in the population, choosing 0.5 for P .

• z = 1.645 (90% confidence), 1.960 (95% confidence), and 2.576 (99% confidence) for d ranging from 0.01 to 0.25, and for P ranging from 0.05 to 0.90 in increments of 0.05.

Sample size

• A district medical officer seeks to estimate the proportion of children in the district receiving appropriate childhood vaccinations. Assuming a simple random sample of a community is to be selected, how many children must be studied if The resulting estimate is to fall within 10 percentage points of the true proportion with 95% confidence?

Sample size

• Using the formula:

• N= 1.96*1.96*0.5*0.5/0.1*0.1= 96.04=97

Sample size

• It should be noted that 97 is the requirement if simple random Sampling is to be used. This would never be the case in an actual field survey.

• As a result, the sample size would go up by the amount

• of the "design effect"!. • For example, if cluster samplingc were to be used,

the design effect might be estimated as 2. • This means that in order to obtain the same precision,

twice as many individuals must be studied with cluster sampling as with the simple random sampling strategy. Hence, 184 subjects would be required.

Sample size

• example, to require the estimate of P to fall within 10% of P rather than to within 10 percentage points of P.

• example, • if the true proportion vaccinated was0.20, the

strategy used in the above example would result in estimates falling between .10 and 0.30 in 95 out of every 100 samples drawn from this population.

• Instead,if we require our estimate to fall within 10% of 0.20, we would fmd that 95 out of every 100 samples would result in estimates between 0.20+0.1(0.20) = 0.22 and 0.20-0.1(0.20) = 0.18.

Sample size for desired precision

e,= the desired precision,

Sample size

• Q determine the sample size necessary to estimate the proportion vaccinated in the population to within 10% of the true value?

Hypothesis testing for a single population proportion

point "c" represents, for the sampling distribution centered at Po (i.e., thedistribution which would result if the null hypothesis were true), the upper lOO(cx)tt percent point of the distribution of p:

and, for the sampling distribution centered at P a (i.e., the distribution which would resultif the alternate hypothesis were true), the lower 1 00(~) tt percent point of the distributionofp:

sample size, for single sample hypothesis testing situation

Sample size

• During a virulent outbreak of neonatal tetanus, health workers wish to determine whether the rate is decreasing after a period during which it had risen to a level of 150 cases per thousand live births. What sample size is necessary to test

• Ho:P=0.15 at the0.05 level if it is desired to have a 90% probability of detecting a rate of 100 per thousand if that were the true proportion?

Using

Sample size

• with Po=0.15, P a=0.1 0, a=0.05, and P 0=0.1 (since the desired power is 90%).

• as Pa gets further and further away from P, the necessary sample size decreases.

sample size for this one-sample, two-sided hypothesis testing

situation,

Sample size

• In determining sample size for this one-sample, two-sided hypothesis testing situation, the problem is that we cannot be sure whether P a was larger than or smaller than P

• Hence, to determine adequate sample size, it is necessary to compute n twice; once with P a larger by a stated amount than P 0 and again with P a less than P by that stated amount.

• The appropriate sample size is the larger of these two numbers.

Sample size

Suppose the success rate for surgical treatment of a particular heart condition is widely reported in the literature to be 0.70. A new medical treatment has been proposed which is alleged to offer equivalent treatment success. A hospital without the necessary surgical facilities or staff has decided to use the new medical treatment on all new patients presenting with this condition. How many patients must be studied to test H :P=0.70 versus Ha:P;e0.70 at the 0.05 level if it is desired to have 90% power of detecting a difference in proportion of success of 1 0 percentage points or greater?

Sample size

Since P a may be less than P by 1 0 percentage points, the computations are performed again using P a =0.6

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