Question

The property tax paid by homeowners in a large city was determined to be normally distributed with a mean of $2,800 and a standard deviation of $400. A random sample of 4 homes was drawn.

a. what is the probability distribution of the mean of the sample of four homes?

b. Determine the probability the sample mean falls between $2,500 and $2,900.

c. How would you answer this question if property tax is not normally distributed?

Answer 1

Solution

Let X = property tax ($) paid by homeowners in a large city.

We are given X ~ N(2800, 400²) ................................................................................................................................... (1)

Back-up Theory

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then,

Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution and hence

P(X ≤ or ≥ t) = P[{(X - µ)/σ} ≤ or ≥ {(t - µ)/σ}] = P[Z ≤ or ≥ {(t - µ)/σ}] .………….................................................……...…(2)

X bar ~ N(µ, σ²/n),……………………………………………….............................................................................….…….(3),

where X bar is average of a sample of size n from population of X.

Probability values for the Standard Normal Variable, Z, can be directly read off from

Standard Normal Tables ………………………………………................................................…………..………………… (2a)

or can be found using Excel Function: Statistical, NORMSDIST(z) which gives P(Z ≤ z) ..........................................…(2b)

Now, to work out the solution,

Part (a)

Vide (3) and (1), probability distribution of the mean of the sample of four homes is:

N(2800, 200²) [200 is the standard deviation] Answer 1

Part (b)

Probability the sample mean falls between $2,500 and $2,900

= P(2500 ≤ Xbar ≤ 2900)

= P[{(2500 - 2800)/200} ≤ Z ≤ {(2900 - 2800)/200}] [vide (2)]

= P(- 1.5 ≤ Z ≤ 0.5)

= P(Z ≤ 0.5) - P(Z ≤- 1.5)

= 0.6915 – 0.0668 [vide (2b)]

= 0.6247 Answer 2

Part (c)

If the distribution of property tax is fairly symmetric, even if not Normal, the above principle can still be applied since by Central Limit Theorem. sample average would be Normal. Answer 3

DONE