Question

The age, in years, of cars at a used car dealership. Select the most likely value of r for this data set.

Option 1: −0.88
Option 2: 0.17
Option 3: 0.56
Option 4: 0.88

Answer 1

The age of cars in a dealership is uniformly distributed between 0.5 and 7.5 years. To find the probability of a car being less than four years old, we divide the interval of interest by the total interval length. The calculation shows a probability of 0.5, and a sketch would show a shaded region on the uniform distribution graph between 0.5 to 4 years.

Step-by-step explanation:

To answer the question regarding the age of cars at a used car dealership, we need to use the concept of probability distribution, specifically uniform distribution, as the ages of cars in the mentioned example range between 0.5 years (6 months) to 9.5 years. Assuming that we are only considering the cars that are less than 7.5 years old, we can calculate the probability of a car being less than four years old.

Step-by-Step Solution:

1. Firstly, identify the interval for the uniform distribution, which in this case is 0.5 years to 9.5 years.
2. Since the question specifies cars less than 7.5 years old, we adjust the interval to 0.5 years to 7.5 years for our calculation.
3. Now, we need to find the probability that a car is less than four years old within this interval. Since it's uniformly distributed, the probability density is constant and equal to 1/(b-a) where 'a' is the minimum age and 'b' is the maximum age for the range of interest.
4. Calculate the probability by dividing the length of the interval of interest (3.5 years, from 0.5 to 4) by the total length of the interval for cars less than 7.5 years old (7 years, from 0.5 to 7.5).
5. Therefore, the probability P(X < 4) = 3.5 / 7 = 0.5.

To sketch the graph, we would draw a rectangle with a base spanning from 0.5 to 7.5 on the x-axis and a height equal to the probability density. The area of interest would be shaded from 0.5 to 4 on this graph.

Considering these steps, the conclusion is that for cars less than 7.5 years old at the dealership, the value of r is not relevant, as r typically denotes a correlation coefficient, which is not applicable in this context of determining probability for a uniformly distributed variable.