The amount of sales tax paid on a purchase is rounded to the nearest cent Assume that the round-off is uniformly distributed in the interval cent a Write the formula for the probability curve describ

a. The probability curve describing the round-off error can be modeled as a uniform distribution with minimum value 0 and maximum value 0.005 (since the round-off can be up to half a cent in either direction). The probability density function is:

f(x) = 1/(0.005-0) = 200 for 0 ≤ x ≤ 0.005 f(x) = 0 otherwise

b. The graph of the probability curve is a rectangle with height 200 and width 0.005, centered at x = 0.0025.

c. The probability of the round-off error exceeding 0.3 cent or being less than 0.005 cent is the sum of the probabilities of these two events:

P(X > 0.003) + P(X < 0.00005) = (0.005 - 0.003) * 200 + 0.00005 * 200 = 0.4%

d. The mean of the round-off error is:

μ = (0 + 0.005)/2 = 0.0025

The variance of the round-off error is:

σ^2 = (0.005-0)^2/12 = 0.00000208

So the standard deviation is:

σ = sqrt(0.00000208) = 0.00144

e. The probability of the round-off error being within one standard deviation of the mean is:

P(|X-μ| < σ) = P(-0.00144 < X-0.0025 < 0.00144) = P(0.00106 < X < 0.00394) = (0.00394-0.00106) * 200 = 0.576%