Problem 42 - Basic Statistics for Business and Economics

Question

Fast Service Truck Lines uses the Ford Super Duty F-750 exclusively. Management made a study of the maintenance costs and determined the number of miles traveled during the year followed the normal distribution. The mean of the distribution was 60,000 miles and the standard deviation 2,000 miles. a. What percent of the Ford Super Duty F-750s logged 65,200 miles or more? b. What percent of the trucks logged more than 57,060 but less than 58,280 miles? c. What percent of the Fords traveled 62,000 miles or less during the year? d. Is it reasonable to conclude that any of the trucks were driven more than 70,000 miles? Explain.

Step-by-Step Solution

Answer

a. 0.47% logged 65,200+ miles. b. 12.41% logged between 57,060 and 58,280 miles. c. 84.13% logged 62,000 miles or less. d. It is not reasonable; probability is nearly zero.

Step 1: Understand the Normal Distribution

The problem states the mileage distribution is normal with a mean (\(\mu\)) of 60,000 miles and a standard deviation (\(\sigma\)) of 2,000 miles. For such normal distributions, percentages can be calculated using the standard normal distribution (Z-distribution).

Step 2: Calculate Z-Score for 65,200 Miles or More (Part a)

To find the percentage of trucks that logged 65,200 miles or more, calculate the Z-score using the formula \( Z = \frac{X - \mu}{\sigma} \). Thus, \( Z = \frac{65,200 - 60,000}{2,000} = 2.6 \). Using the standard normal distribution table, find the probability corresponding to \( Z = 2.6 \). Look for \( Z = 2.6 \) in the table, which gives approximately 0.9953. Since the table gives cumulative from the left, subtract from 1: 1 - 0.9953 = 0.0047. Thus, 0.47% of the trucks logged 65,200 miles or more.

Step 3: Calculate Z-Scores for 57,060 and 58,280 Miles (Part b)

Find the Z-score for 57,060 miles: \( Z = \frac{57,060 - 60,000}{2,000} = -1.47 \). Look up this Z-score in the table for probability, which is approximately 0.0708. Then, find the Z-score for 58,280 miles: \( Z = \frac{58,280 - 60,000}{2,000} = -0.86 \). This corresponds to approximately 0.1949 in the table. Subtract the probability of 57,060 miles from that of 58,280 miles: 0.1949 - 0.0708 = 0.1241. Thus, 12.41% of trucks logged between these miles.

Step 4: Calculate Z-Score for 62,000 Miles or Less (Part c)

Calculate the Z-score for 62,000 miles: \( Z = \frac{62,000 - 60,000}{2,000} = 1.0 \). Using the standard normal table, the probability for \( Z = 1.0 \) is approximately 0.8413. Thus, 84.13% of the trucks logged 62,000 miles or less.

Step 5: Evaluate Reasonability for Miles Over 70,000 (Part d)

Calculate Z-score for 70,000 miles: \( Z = \frac{70,000 - 60,000}{2,000} = 5\). The Z-score of 5 is extremely rare in normal distribution, meaning the probability is almost 0 (can confirm by checking the table). This suggests it is highly improbable that any truck exceeded 70,000 miles, thus it is not reasonable to expect this to happen.

Key Concepts

Z-score calculation

In statistics, calculating the Z-score is essential when working with the normal distribution. The Z-score allows us to determine how far away a specific data point is from the mean. To find a Z-score, you use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is the value you're examining
• \( \mu \) is the mean of the distribution
• \( \sigma \) is the standard deviation

The Z-score tells us how many standard deviations away from the mean our value is.
For example, in the exercise, to figure out the Z-score of trucks logging 65,200 miles, the calculation becomes \[ Z = \frac{65,200 - 60,000}{2,000} = 2.6 \].This indicates that 65,200 miles is 2.6 standard deviations above the mean mileage. This calculation helps us find related probabilities using the normal distribution table.

Standard deviation

Standard deviation is a metric used to quantify the amount of variation or dispersion in a set of values. It's an integral part of many statistical analyses, particularly when dealing with the normal distribution.
The formula for calculating the standard deviation in a dataset is designed to give you a measure of how spread out the data points are around the mean. In practical terms:

• If the standard deviation is small, the data points tend to be close to the mean.
• If the standard deviation is large, the data points are spread out over a wider range.

In the exercise involving truck mileage, the standard deviation is applied to understand how mileage varies among trucks. A standard deviation of 2,000 miles indicates how truck mileages deviate from the 60,000-mile average.

Probability in statistics

Probability in statistics is the likelihood or chance of an event occurring. It's a fundamental concept that underpins many statistical methods and analyses. When looking at normal distributions, probability helps us understand how data is distributed around the mean.
When calculating probabilities related to the normal distribution:

• We use the Z-score to represent a point within the distribution.
• The Z-score is then used to find the corresponding probability in a standard normal distribution table.

For instance, in the exercise, to determine the probability of trucks traveling more than 65,200 miles, we computed a Z-score of 2.6.
Next, we looked up this Z-score in the table to determine the probability that trucks would travel this distance or more.

Normal distribution table

A normal distribution table, also known as a Z-table, is a valuable tool in statistics. It helps us find the probability that a statistic is observed below, above, or between values on a standard normal distribution.
The table is used as a reference for Z-scores:

• It is structured to show the cumulative probability of a value being below a given Z-score.
• By subtracting these probabilities from 1, we can determine the likelihood of being above a specific Z-score.

In the exercise, after calculating the Z-score for trucks logging 65,200 miles (which was 2.6), we used the normal distribution table to find a cumulative probability of approximately 0.9953.
Subtracting this from 1 gave us the percentage of trucks exceeding 65,200 miles, helping facilitate further analysis.