A construction zone on Kings Road has a posted speed of 45 kilometers per hour. The speeds of vehicle passing through this construction zone are normally distributed with a mean of 46 kilometers per hour and a standard deviation of 4 kilometers per hour. Find the percentage of vehicles passing through this construction that are
i) Exceeding the posted speed limit.
ii) Traveling at speeds between 50 and 55 kilometers per hour
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Multiply by 100.
42
To find the percentage of vehicles passing through the construction zone that are:
i) Exceeding the posted speed limit:
To calculate this, we need to find the area under the curve to the right of the posted speed limit.
Step 1: Standardize the posted speed limit using the z-score formula.
z = (x - μ) / σ
where x is the value we want to standardize (45 km/h), μ is the mean (46 km/h), and σ is the standard deviation (4 km/h).
z = (45 - 46) / 4
z = -1 / 4
z = -0.25
Step 2: Find the area to the right of the standardized value using a Z-table or calculator. The Z-table gives us the area to the left, so we subtract this from 1 to get the area to the right.
P(z > -0.25) = 1 - P(z < -0.25)
Step 3: Look up the value in the Z-table or use a calculator to find the corresponding probability.
Using a Z-table or calculator, we find that P(z < -0.25) is approximately 0.4013.
Step 4: Calculate the percentage of vehicles exceeding the posted speed limit.
P(z > -0.25) = 1 - 0.4013 = 0.5987
Therefore, approximately 59.87% of the vehicles passing through the construction zone are exceeding the posted speed limit.
ii) Traveling at speeds between 50 and 55 kilometers per hour:
To calculate this, we need to find the area under the curve between the speeds of 50 and 55 km/h.
Step 1: Standardize the lower and upper speeds using the z-score formula.
For the lower speed (50 km/h):
z1 = (x1 - μ) / σ = (50 - 46) / 4 = 1
For the upper speed (55 km/h):
z2 = (x2 - μ) / σ = (55 - 46) / 4 = 2.25
Step 2: Find the area between the standardized values using the Z-table or calculator.
P(1 < z < 2.25)
Step 3: Look up the values in the Z-table or use a calculator to find the corresponding probabilities.
Using a Z-table or calculator, we find that P(z < 1) is approximately 0.8413 and P(z < 2.25) is approximately 0.9878.
Step 4: Calculate the percentage of vehicles traveling at speeds between 50 and 55 km/h.
P(1 < z < 2.25) = P(z < 2.25) - P(z < 1)
P(1 < z < 2.25) = 0.9878 - 0.8413 = 0.1465
Therefore, approximately 14.65% of the vehicles passing through the construction zone are traveling at speeds between 50 and 55 kilometers per hour.