Question

Carbon monoxide (CO) emissions for cars vary with mean 2.9 gm/mi and standard deviation 0.4 gm/mi. The Ponaxtiruchiranalignim Car Rental Fleet has 10 of these cars in its fleet, acquired from various (i.e., random) sources. The CO emissions follow a normal distribution.

a) What is the probability that a randomly selected car from the street has a CO emission greater than 3.1 gm/mi.
b) What is the probability that a randomly selected car from the fleet has CO emissions in excess of 3.1 gm/mi.
c) What is the shape of the distribution of the CO emissions of the cars in the fleet?
d) Construct the normal curve for the cars in the fleet.
e) What are the conditions for the Central Limit

Answer 1

The probability of a car having a CO emission greater than 3.1 gm/mi can be found using the z-score and standard normal distribution. The distribution of CO emissions for the fleet is normally distributed. The Central Limit Theorem applies with the conditions that samples be independent and of a sufficient size.

Step-by-step explanation:

When dealing with normally distributed data, such as carbon monoxide (CO) emissions for cars, there are several key tasks you can perform:

To find the probability that a randomly selected car has a CO emission greater than 3.1 gm/mi, calculate the z-score and then refer to the standard normal distribution table or use statistical software. The z-score is calculated as follows: z = (X - μ) / σ, where X is the value in question (3.1 gm/mi), μ is the mean (2.9 gm/mi), and σ is the standard deviation (0.4 gm/mi).

The probability for a single car from the fleet is the same as above since it is still one observation from the same normal distribution.

The distribution of the CO emissions for the fleet will also be normally distributed because the original population of CO emissions is normally distributed.

You can draw a normal curve by plotting the mean at the center of the horizontal axis and marking points at intervals of the standard deviation.

Concerning the Central Limit Theorem, it has two major conditions: the samples must be independent, and the sample size should be sufficiently large (usually n > 30).

Answer 2

• a. The probability that a randomly selected car from the street has a CO emission greater than 3.1 gm/mi is approximately 0.6915.
• b. The probability that a randomly selected car from the fleet has CO emissions in excess of 3.1 gm/mi is 0.6915.
• c.The shape of the distribution of the CO emissions of the cars in the fleet is approximately normal.
• d.The normal curve for the cars in the fleet is a bell-shaped curve, with the mean at the center and the standard deviation determining the spread of the curve.
• e . The conditions for the Central Limit Theorem to apply are: The sample should be randomly selected or obtained through a random process, The sample size should be sufficiently large and The individual observations in the sample should be independent of each other.

Step-by-step explanation:

a) To find the probability that a randomly selected car from the street has a CO emission greater than 3.1 gm/mi, we need to calculate the area under the normal distribution curve to the right of 3.1.

First, we need to standardize the value of 3.1 using the mean and standard deviation given. The standardized value, also known as the z-score, is calculated using the formula:

z = (x - μ) / σ

where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.

In this case, the mean is 2.9 gm/mi and the standard deviation is 0.4 gm/mi.

z = (3.1 - 2.9) / 0.4

z = 0.2 / 0.4

z = 0.5

Now, we can use a standard normal distribution table or a calculator to find the probability associated with a z-score of 0.5.

Using the table or a calculator, we find that the probability is approximately 0.6915.

Therefore, the probability that a randomly selected car from the street has a CO emission greater than 3.1 gm/mi is approximately 0.6915.

b) To find the probability that a randomly selected car from the fleet has CO emissions in excess of 3.1 gm/mi, we use the same process as in part (a). Since the fleet consists of 10 cars, the probability is the same as in part (a), which is approximately 0.6915.

c) This is because the CO emissions follow a normal distribution and are characterized by a mean and a standard deviation.

d) The curve is symmetrical around the mean, and the area under the curve represents the probabilities of different CO emission values.

e) The conditions for the Central Limit Theorem to apply are:

• 1.The sample ought to be chosen at random or acquired using a random procedure.
• 2. A sizable enough sample size is required. A standard recommendation is that there should be a minimum of 30 participants in the sample.
• 3. Every observation in the sample ought to stand alone from the others. This implies that the value of one observation in the sample shouldn't affect the value of another observation.