2. Donations to tax-exempt organizations (e.g. Red Cross) not only go to the sta

Question

2. Donations to tax-exempt organizations (e.g. Red Cross) not only go to the stated charitable purpose, but are also used to cover fundraising expenses and overhead. The “charitable commitment” is the “cents per dollar” of donations that goes toward the charitable purpose. A random sample of 50 charitable organizations was selected and their charitable commitments recorded. The sample found a mean charitable commitment of 75.3 cents per dollar, and a standard deviation of 12.8 cents per dollar.

   (a)   Estimate with 95% confidence the true mean charitable commitment among all charitable organizations.

(b)   How many charitable organizations would need to be sampled to estimate the mean to within a margin of error of 2 cents per dollar, at 95% confidence?

Explanation / Answer

n =50 , mean = 75.3 , s = 12.8

a) z value at 95% CI = 1.96

CI = mean + / - z * ( s/sqrt(n))

= 75.3 + / - 1.96 * ( 12.8 / sqrt(50))

= (71.75 , 78. 85)

b) ME = 2 , z value = 1.96 , s = 12.8

ME = z * ( s / sqrt(n))

n = (z * s / ME)^2

= ( 1.96 * 12.8 / 2)^2

= 157.35 = 157

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