Question

The NDP propose a trial tax-increase program. Under this trial the social-insurance numbers (SINs) of all super-rich Ontario residents are placed in a barrel, and a random sample of 1200 of them is selected (assume there are a total of 6000 super-rich Ontario residents) for taxation at the higher rate. The tax-income earned from this program will depend on the total income earned by the 1200 sampled residents. (note: that if the sample consisted of just two individuals, with earnings of say $900,000 and $630,000 then the total income would be $1,530,000)

The NDP recognize that this total income is a random variable and have asked you to determine some summaries of this random variable as specfied below.

(b) (2 points) Its expected value

(c) (2 points) Its standard deviation

(d) (4 points) Its inter-quartile range (IQR)

Answer 1

Answer:

Given that,

(a).

The NDP propose a trial tax-increase program. Under this trial the social-insurance numbers (SINs) of all super-rich Ontario residents are placed in a barrel, and a random sample of 1200 of them is selected (assume there are a total of 6000 super-rich Ontario residents) for taxation at the higher rate.

The tax-income earned from this program will depend on the total income earned by the 1200 sampled residents.

(Note: that if the sample consisted of just two individuals, with earnings of say $900,000 and $630,000 then the total income would be $1,530,000)

The NDP recognize that this total income is a random variable and have asked you to determine some summaries of this random variable as specfied below.

Let X be the earnings of super rich people.

be the expected earning of super rich people.

is the standard deviation earnings of super rich people.

Assume X N(, )

Here, given that, =$ 750,000/yr and =$ 400,000/ yr

Then,

X N(=750,000, =(400,000)^2)

Now, The probability that earnings of super rich people excess $800000/yr.

=P(X > 800000)

=1-P(X 800000)

Since,

[ be the cdf of standard normal distribution, It's value calcualted form normal distribution table].

=1-P(Z 0.125)

=1-(0.125)

=1-0.54974

=0.45026

(b).

Its expected value:

Now we know,

Suppose X1, ...........,Xn N(, )

Then, the sample mean,

Here,

The sample size, n=1200

=$ 750,000/yr and =$ 400,000/ yr

So, here

So, the expected value of sample mean,

(c).

Its standard deviation:

The standard deviation of sample mean,

(d).

Its inter-quartile range (IQR):

Suppose the percentile=Q3 and 25th percentile=Q1

Then, we know

IQR=Q3-Q1

Now,

Since,

We know from normal distribution table.

P(Z 0.675)=0.75

Similarly for 25th percentile, we have,

And, we know

So, interquartile range,

IQR=Q3-Q1

=757794.2886-742205.7714

=15588.45723