Question

1. The nitrous oxide (N2O) emission level of all diesel cars of a particular model has a true (population) mean of 25 (micrograms/km) and std deviation of 2. Assume that the population distribution is Normal. A random sample of these cars is taken.

a) find the probability that the sample mean emisions will be less than 23 mg/km if the sample is of (i) one, (ii) four, (iii) 100, cars.

b) explain why the three answers differ, and illustrate with a graph.

c) what difference would it make if you didn’t assume that the population distribution is Normal?

Answer 1

1. The nitrous oxide (N2O) emission level of all diesel cars of a particular model has a true (population) mean of 25 (micrograms/km) and std deviation of 2. Assume that the population distribution is Normal. A random sample of these cars is taken.

We have given,

= 25, = 2

a)-(i) find the probability that the sample mean emisions will be less than 23 mg/km if the sample is of 1 (i.e. n=1)

That is

= p(Z < -1)

= NORMSDIST(-1)

  = 0.1587

a)-(ii) find the probability that the sample mean emisions will be less than 23 mg/km if the sample is of 4 (i.e. n=4)

That is

= p(Z < -2)

= NORMSDIST(-2)

  = 0.0228

a)-(iii) find the probability that the sample mean emisions will be less than 23 mg/km if the sample is of 100 (i.e. n=100)

That is

= p(Z < -10)

= NORMSDIST(-10)

  = 0.000

b) explain why the three answers differ, and illustrate with a graph.

Since as the sample size n increases, standard deviation of mean (), decreases resulting probability decreases.

c) what difference would it make if you didn’t assume that the population distribution is Normal?

As per the Central Limit Theorem, sampling distribution of sample mean () follows normal distribution regardless of what distribution X follows. Hence there won't be any difference though we have not assumed normal distribution.