Final answer:
The probability that the loan is greater than $49,000 is 0.1587. The probability that the loan is less than $43,000 is 0.3085. The probability that the loan falls between $41,000 and $50,200 is 0.7445.
Step-by-step explanation:
(a) The loan is greater than $49,000:
To find this probability, we need to calculate the z-score and then find the area under the normal curve to the right of the z-score.
First, we calculate the z-score using the formula:
z = (x - mean) / standard deviation
Plugging in the values, we have:
z = (49000 - 45000) / 4000 = 1
Next, we look up the area to the right of z = 1 in the standard normal distribution table. The area is 0.1587.
Therefore, the probability that the loan is greater than $49,000 is 0.1587.
(b) The loan is less than $43,000:
To find this probability, we again calculate the z-score:
z = (43000 - 45000) / 4000 = -0.5
Then, we look up the area to the left of z = -0.5 in the standard normal distribution table. The area is 0.3085.
Therefore, the probability that the loan is less than $43,000 is 0.3085.
(c) The loan falls between $41,000 and $50,200:
To find this probability, we calculate the z-scores for both values:
z1 = (41000 - 45000) / 4000 = -1
z2 = (50200 - 45000) / 4000 = 1.3
Next, we find the area to the left of z = -1 and to the left of z = 1.3 in the standard normal distribution table. The areas are 0.1587 and 0.9032, respectively.
To find the area between these two z-scores, we subtract the area to the left of z = -1 from the area to the left of z = 1.3:
Area = 0.9032 - 0.1587 = 0.7445.
Therefore, the probability that the loan falls between $41,000 and $50,200 is 0.7445.