Step-by-step explanation:
To estimate the required sample size in both cases, we can use the formula:
n = (Z * σ / E)^2
where:
n = required sample size
Z = z-value (which corresponds to a 95% confidence level)
σ = standard deviation in speed
E = margin of error (limit of acceptable error)
a) For a 95% confidence level, the z-value is 1.96 (You can find this value from the z-score table). Given that the standard deviation in speed is ±4.5mi/hr and the margin of error is ±1.3mi/hr:
n = (1.96 * 4.5 / 1.3)^2
n ≈ (6.804 / 1.3)^2
n ≈ 5.233^2
n ≈ 27.38
Since we can't have a fraction of a sample or a negative sample, the Engineer should collect at least 28 spot speeds to estimate the average speed on the roadway at a 95% confidence level with a margin of error of ±1.3mi/hr.
b) If the margin of error should be within ±1mi/hr, we can adjust the formula with the new margin of error:
n = (1.96 * 4.5 / 1)^2
n ≈ (6.804 / 1)^2
n ≈ 46.28
For the estimate of the average to be within ±1mi/hr, the Engineer should collect at least 47 spot speeds at a 95% confidence level.