Question

a random sample of 100,000 credit sales in a department store showed an average sale of $87.25. from past data, it is known that the standard deviation of the population is $20.00. with a .95 probability, determine the margin of error.

Answer 1

The margin of error for the probability is 0.124

The binomial distribution is widely used to simulate the margin of error and number of successes in a large number of observations n drawn under replacement from a single population of size n which can be used to define the margin of error based on the confidence level.

If sampling is done without replacement, drawings are not independent, hence a hypergeometric model rather than a binomial one is utilized as the generative model. However, the binomial distribution is still a good approximation and is routinely used for N much larger than n.

Given n = 100000

Standard deviation (σ)= 20

Confidence level (α) = 0.95

So (1-α) = 1 - 0.95 = 0.05

∴Critical value = Z(α/2) = Z₀.₀₂₅ = 1.96

Calculating the margin of error:

Margin of error

$=Z_{\alpha /2}\times \frac{\rho}{\sqrt n}\\\\1.96 \times \frac{20}{\sqrt100000} \\\\=0.1239..\\\\\approx 0.124$

Hence the margin of error for the probability is 0.124 .

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