3. Suppose you work for the Ford Motor Company and must establish regulations concerning the maximum number of people who can ride in a minivan. Ford tells you that the total number of people in a minivan follows a normal distribution with a mean of 6 and a standard deviation of 2.
a. What is the probability that the total number of people in the van is between 2 and 4 people?
b. What is the probability that the total number of people in the van is between 5 and 7 people?
c. What is the probability that the total number of people in the van is greater than 10 people?
Here the distribution ~N(6,2²), i.e. mean=6, variance = 2²=4.
Standardize the given variable to Z and look up probabilities from the normal distribution table.
To answer these questions, we need to use the normal distribution and its properties. We know that the total number of people in a minivan follows a normal distribution with a mean of 6 and a standard deviation of 2.
To find probabilities in a normal distribution, we can use the z-score formula:
z = (x - μ) / σ
Where:
- x is the value we want to find the probability for.
- μ is the mean of the distribution.
- σ is the standard deviation of the distribution.
Once we have the z-score, we can refer to the standard normal distribution table or use statistical software to find the corresponding probability.
a. To find the probability that the total number of people in the van is between 2 and 4 people:
First, we need to calculate the z-scores for both values.
For x = 2:
z = (2 - 6) / 2 = -4 / 2 = -2
For x = 4:
z = (4 - 6) / 2 = -2 / 2 = -1
Using the standard normal distribution table or software, we can find the probabilities associated with these z-scores.
b. Similarly, to find the probability that the total number of people in the van is between 5 and 7 people, we need to calculate the z-scores for x = 5 and x = 7, and then find the corresponding probabilities.
For x = 5:
z = (5 - 6) / 2 = -1 / 2 = -0.5
For x = 7:
z = (7 - 6) / 2 = 1 / 2 = 0.5
c. Finally, to find the probability that the total number of people in the van is greater than 10 people, we can calculate the z-score for x = 10 and find the corresponding probability.
For x = 10:
z = (10 - 6) / 2 = 4 / 2 = 2
Using these z-scores, we can find the probabilities associated with each situation. Remember to refer to the standard normal distribution table or use statistical software.