₹418.7

Step-by-step explanation:

To solve this problem, we can use the properties of the normal distribution and standard scores (also known as z-scores).

Given:

Mean (μ) = ₹520

Standard Deviation (σ) = ₹60

(i) To find the number of persons having incomes between ₹400 and ₹550, we need to calculate the proportion of the population within this range.

First, we convert the given incomes into z-scores using the formula:

z = (x - μ) / σ

For ₹400:

z₁ = (400 - 520) / 60 = -2

For ₹550:

z₂ = (550 - 520) / 60 = 0.5

Using a standard normal distribution table or a calculator, we can find the corresponding probabilities for these z-scores.

P(-2 < z < 0.5) = P(z < 0.5) - P(z < -2)

Next, we look up the values from the standard normal distribution table or use a calculator to find the corresponding probabilities:

P(z < 0.5) ≈ 0.6915

P(z < -2) ≈ 0.0228

Therefore,

P(-2 < z < 0.5) ≈ 0.6915 - 0.0228 = 0.6687

To find the number of persons within this income range, we multiply this proportion by the total number of persons:

Number of persons = Proportion * Total number of persons

Number of persons = 0.6687 * 10,000

Number of persons ≈ 6687

So, approximately 6687 persons have incomes between ₹400 and ₹550.

(ii) To find the lowest income of the richest 500, we need to calculate the z-score corresponding to the 500th person from the highest income end.

To find the z-score, we use the formula:

z = (x - μ) / σ

For the lowest income of the richest 500, we want the 500th z-score from the highest end, which can be calculated using:

z = -1.645 (approximately)

Now, we can convert the z-score back into the corresponding income value:

x = z * σ + μ

x = -1.645 * 60 + 520

x ≈ ₹418.7

Therefore, the lowest income of the richest 500 individuals is approximately ₹418.7.