Question

1) Speed data collected on an urban road way yielded a standard deviation in speed of ± 4.5mi/hr. a) If an Engineer wishes to establish to estimate the average speed on the roadway at 95% confidence level so that the estimate is ± 1.3mi/hr. of limit of acceptable error, how many spot speeds should be collected? b) If the estimate of the average must be within ± 1mi /hr., what should the sample size be?​

Answer 1

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.