the customer accounts at a certain departmental store have an average balance of Rs 480 and a standard deviation of Rs 160 assuming that the account balances are normally distributed what is the proportion of the accounts is over 600 and what proportion of the accounts is between Rs 240 and Rs 360?
Answers
The proportion of the accounts that is over 600 is 22.66% and the proportion of the accounts that is between Rs 240 and Rs 360 is 15.98%.
Given: The customer accounts at a certain departmental store have an average balance of Rs 480 and a standard deviation of Rs 160.
To find: What is the proportion of the accounts is over Rs. 600 and what proportion of the accounts is between Rs 240 and Rs 360.
Solution:
We are said to assume that the account balances are normally distributed.
Let a random variable 'x' be the balance of the customer accounts. Variable x is normally distributed with μ = 480 and б = 160 and the standard deviation 'z' is:
z = (x – μ)/б = (x – 480)/160
Now,
If x = 600,
z = (x – 480)/160 = (600 – 480)/160 = 120/160 = 0.75
Thus, P(x > 600) = P(z > 0.75) = 0.5 – 0.2734 = 0.2266
Therefore, 22.66% of the accounts have a balance that is over Rs. 600.
Again,
Proportion of the accounts is between Rs. 240 and Rs. 360 is given by P(240 ≤ x ≤ 360).
If x = 240, then z = (x – 480)/160 = (240 – 480)/160 = – 240/160 = – 1.5
If x = 360, then Z = (x – 480)/160 = (360 – 480)/160 = = – 120/160 = – 0.75
Thus, P(240 ≤ x ≤ 360) = P(– 1.5 ≤ z ≤ – 0.75) = 0.5 – 0.3402 = 0.1598
Therefore, 15.98% of the accounts are between Rs 240 and Rs 360.
Therefore, 22.66% of the accounts have a balance that is over Rs. 600 and 15.98% of the accounts is between Rs 240 and Rs 360.
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