Sales Tax Round-Off Error: Uniform Distribution Analysis

a. The probability curve describing the round-off error is a uniform distribution with a minimum value of -0.005 cent and a maximum value of 0.005 cent. The formula for the probability density function (PDF) of a uniform distribution is:

f(x) = 1/(b-a) for a ≤ x ≤ b

where a is the minimum value, b is the maximum value, and f(x) is the probability density function.

In this case, a = -0.005 and b = 0.005, so the PDF is:

f(x) = 1/(0.01) for -0.005 ≤ x ≤ 0.005

b. The graph of the probability curve describing the round-off error is a rectangle with a height of 100 (to represent a percentage) and a width of 0.01 (to represent the range of values from -0.005 to 0.005). The area of the rectangle is 1, representing the total probability.

c. To find the probability that the round-off error exceeds 0.3 cent or is less than -0.3 cent, we need to find the area under the curve outside the interval [-0.003, 0.003]. This can be done by finding the area of the two triangles outside the interval:

P(round-off error > 0.3 cent or < -0.3 cent) = 2(0.5)(0.0025) = 0.0025

d. The mean of the round-off error is zero, since the distribution is symmetric around zero. The standard deviation can be found using the formula for the standard deviation of a uniform distribution:

σ = (b-a)/√12

where σ is the standard deviation, a is the minimum value, and b is the maximum value.

In this case, a = -0.005 and b = 0.005, so:

σ = (0.005 - (-0.005))/√12 ≈ 0.0014 cent

e. To find the probability that the round-off error will be within one standard deviation of the mean, we need to find the area under the curve between -0.0014 and 0.0014 cent. This can be done by finding the area of the rectangle between these values:

P(-0.0014 ≤ round-off error ≤ 0.0014) = 100(0.0028) = 0.28

Therefore, there is a 28% chance that the round-off error will be within one standard deviation of the mean.