Estimating the Population Proportion

Estimating the Proportion:

Here we are estimating the population proportion, p.

All the estimation done here is mainly based on the fact that the normal can be employed to approximate the binomial distribution when both np and nq are at least 5. Therefore, the p that we are talking about is the probability of success on single trial from binomial experiments.

Note:

μ = np
σ = √npq
z = (x - μ)/σ = (x - np)/√npq

The finest point estimate for p is p hat, that is, the sample proportion:

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When the formula for z is divided by n in both numerator and denominator, then the formula for z is:

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Solving this for p to come up with a confidence interval provides the maximum error of estimate as:

E = Zα/2 (√pq/n)

This is not the formula which we will use. The problem with estimation is that you do not know the value of parameter (in this case p), therefore you cannot use it to estimate it - if you knew it, and then there would be no problem to work-out. Therefore we will substitute the parameter by statistic in the formula for maximum error of estimate.

The maximum error of estimate is given by the formula for E which is as shown below. The ‘Z’ here is the z-score received from the normal table, or the bottom of t-table as described in the introduction of estimation. z-score is a factor of the level of confidence; therefore you might get in the habit of writing it later to the level of confidence.

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Whenever we are computing E, it is recommended that you find the sample proportion, p hat, and save it to P on calculator. In this way, we can find q as (1-p). Do not round the value for p hat and employ the rounded value in calculations. This will definitely lead to error. Once we have calculated E, it is recommended to save it to the memory on calculator. On TI-82, an excellent choice would be the letter E. The main reason for this is that, limits for the confidence interval are now found by adding and subtracting the maximum error of estimate from or to the sample proportion.

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