When a randomly selected number A is rounded off to the nearest integer R_A, it is reasonable to assume that the round-off error A - R_A is uniformly distributed on the interval (-.5,.5) If 50 numbers are rounded off to the nearest integer and then averaged, approximate the probability that the resulting average differs from the exact average of the 50 numbers by more than 0.1 (round your answer to 4 digits). 0.0142

Check Explanation.

Step-by-step explanation:

Note that the sample size given in the question is fifty (50).

The round-off error is then uniformly distributed on the interval (-.5,.5). Which makes us to have a uniform distribution parameter a = - 0.5 and b = 0.5.

Step one: look for the mean.

The mean, μ = (a + b) / 2 = ( - 0.5 + 0.5) / 2 = 0. Which means that our mean is zero (0).

Step two: Calculate the standard deviation from the formula below;

Standard deviation,σ = (b - a) / √12.

Standard deviation,σ = (0.5 + 0.5) / √12.

Standard deviation,σ = 0.2887.

Step three: Calculate the standard error.

standard error = σ/√n.

standard error = 0.0408

Therefore, the probability, P;

Probability, P = 1 - P ( - 0.1 < X < 0.1) = 1 - P ((-0.1 - 0) / 0.041) < Z < (0.1-0) / 0.041) = 1 - P (-2.45 < Z < 2.45) = 1 - (0.9929 - 0.0071) = 0.0142.