To find a confidence interval you must first find the margin of error. The equation for margin of error is:
E = Z * sqrt((pq)/n)
Where Z is the critical value, p is the sampled proportion, q is one minus the sample proportion, and n is the number of elements in the sample. We know that p is .65. Since q is 1 - p, q must be .35. And n is given to us as 100.
Z will be a little harder to find. The critical value is z-score value that separates the area that is within the critical level from the area outside the critical level. There will be a positive and a negative z-score that do this, but the critical value is always the positive one. To find the critical value we must first find the area on a normal curve that the critical value will correspond to. To find this value use the equation:
V = ((1 - C) / 2) + C
Where C is the confidence level. By plugging .9 in for C we get:
V = ((1 - .9) / 2) + .9 = .95
Now we must find the z-score that corresponds to this value. Use a calculator or a table to find it. Using my calculator I got approximately 1.64. This is your critical value. Use it in the margin of error equation.
Now that we have everything we need, we can now find the margin of error:
E = (1.64) * sqrt((.65 * .35) / 100) = .0782
Now that we have the margin for error we can calculate the confidence interval. The confidence interval is calculated by the following equation:
p - E %26lt; A %26lt; p + E
Where p is the sample proportion, E is the margin of error, and A is the actual population proportion. Plug in our numbers that we got:
.65 - .0782 %26lt; A %26lt; .65 + .0782
.5718 %26lt; A %26lt; .7282
So we are 90% confident that the population proportion of tax returns that have errors is somewhere between 57.18% and 72.82%.
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