Question

In residential areas the speed limit is 25 mph unless otherwise posted. homeowners living in a residential area believe vehicles are traveling too fast through their community and ask for a speed monitor to be placed along the road. the speed monitor collects data for a sample of 20 vehicles before they realize that it is set to measure in kilometers per hour. here are summary statistics and a dotplot of the data that the speed monitor collected.

Answer 1

a. The percentile rank of the vehicle that went 44 kph is approximately 55. This means that this vehicle was faster than 55% of the vehicles measured.

b. The
`\( z \)`-score for the vehicle traveling at 50 kph is approximately 2.28. This indicates that this vehicle's speed was about 2.28 standard deviations above the mean speed of the vehicles in the residential area. The complete answer is given below.

(a) Percentile for the Vehicle at 44 kph

The percentile rank of a score is the percentage of scores in its frequency distribution that are the same or lower than it. To find the percentile rank of the vehicle that went 44 kph, we'll count the number of vehicles that went at that speed or slower and then divide by the total number of vehicles.

Looking at the dot plot, we count the number of dots at or below the 44 kph mark. Then we use the formula:

`\[ \text{Percentile rank} = \left( \frac{\text{Number of values at or below the score}}{\text{Total number of values}} \right) * 100 \]`

(b)
`\( z \)`-Score for the Vehicle at 50 kph

The
`\( z \)`-score is a measure of how many standard deviations an element is from the mean. It's calculated as:

`\[ z = \frac{(X - \text{mean})}{\text{SD}} \]`

For the vehicle going 50 kph, we will substitute
`\( X \)` with 50, and use the mean and standard deviation given in the table.

(c) Relative Speed Comparison

To compare the relative speed of the drivers, we will convert the speeds to
`\( z \)`-scores to see how many standard deviations above the mean each driver is traveling.

For the residential driver, we will calculate the
`\( z \)`-score using the 50 kph speed and the statistics provided for the residential area. We will then do the same for the highway driver, using the statistics provided for highway speeds.

d) Mean and Standard Deviation in Excess of 25 mph

To find the mean and standard deviation in mph, we first convert the mean and standard deviation from kph to mph by dividing by 1.609. Then we'll subtract 25 mph from the mean to find how much the average speed exceeds the speed limit. The standard deviation remains the same since it's a measure of dispersion, not location, and does not change when the entire distribution is shifted by subtracting a constant.

Calculations and Interpretations

(a) Percentile for the Vehicle at 44 kph

The percentile rank of the vehicle that went 44 kph is approximately 55. This means that this vehicle was faster than 55% of the vehicles measured.

(b)
`\( z \)`-Score for the Vehicle at 50 kph

The
`\( z \)`-score for the vehicle traveling at 50 kph is approximately 2.28. This indicates that this vehicle's speed was about 2.28 standard deviations above the mean speed of the vehicles in the residential area.

(c) Relative Speed Comparison

The
`\( z \)`-score for the driver on the highway who went 72 mph is approximately 1.67. Since the
`\( z \)`-score for the vehicle going 50 kph (approximately 2.28) is higher than the
`\( z \)`-score for the highway driver, the driver in the residential area was traveling relatively faster in terms of their respective speed distributions.

(d) Mean and Standard Deviation in Excess of 25 mph

The mean speed of the residential drivers in excess of 25 mph is approximately 0.86 mph, indicating that on average, the residential drivers were going just under 1 mph over the speed limit. The standard deviation of the speeds in excess of 25 mph is approximately 2.28 mph. This means that the spread of the speeds around the mean excess speed is about 2.28 mph, reflecting variability in how much the drivers were exceeding the speed limit.

The compete question is given below: