Question

[20 pts] Carbon monoxide (CO) emissions for cars vary with mean 2.9 gm/mi and standard deviation 0.4 gm/mi. The Penaxtiruchicanalignim 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 Theorem?

Answer 1

Define random variable X: CO emissions

X follows normal distribution with mean = $\mu$ = 2.9 and standard deviation = $\sigma$ = 0.4

n = sample size = 10

a)

Here we have to find P(X > 3.1)

$P(X > 3.1)=P(\frac{X-\mu}{\sigma}>\frac{3.1-\mu}{\sigma})$

  $=P(z>\frac{3.1-2.9}{0.4})$ where z is standard normal variable

0.2 = P(Z > 0.4

  $=P(z>0.5)$

= 1 - P(z < 0.5)

= 1 - 0.6915 (From statsitcial table of z values

= 0.3085

Probability that a randomly selected car from the street has a CO emission greater than 3.1 gm/min is 0.3085.

b)
Here we have to find P(X > 3.1)

It is same as found in part a

Probability that a randomly selected car from the street has a CO emission in excess of 3.1 gm/min is 0.3085.

c)
Shape of distribution of the CO emissions of the cars in the fleet is bell-shaped, symmetric.

d)

=2.9

e)
Central limit theorem:

If distribution of random variable X is normal then the sampling distribution of sample mean is also normally distributed with mean = $\mu_{\bar{x}}=\mu$ and standard deviation is

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$

If Sample size n is large (n >30) irrespective of distribution of random variable X then the sampling distribution of sample mean is approximately normally distributed with mean = $\mu_{\bar{x}}=\mu$ and standard deviation is

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$