A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 200 and 275. Round to four decimal places.

The probability of a rating that is between 200 and 275 is,

Since the credit ratings are normally distributed.

Now,  apply the formula for normal distribution . Which is shown below,

Where  x  represent credit ratings for applicants  , µ  is  mean  and σ  is  standard deviation.

From the given information,  µ = 200 and   σ = 50

For x = 200,

In normal distribution table, the probability corresponding to the zero z score is 0.5 .

For x = 275,

In normal distribution table, the probability corresponding to the z score of 1.5  is 0.9332

Thus,    

Hence, The probability of a rating that is between 200 and 275 is,

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P(200 ≤ x ≤ 275) = 0.4332

Step-by-step explanation:

Since the credit ratings are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = credit ratings for applicants

µ = mean

σ = standard deviation

From the information given,

µ = 200

σ = 50

The probability of a rating that is between 200 and 275 is expressed as

P(200 ≤ x ≤ 275)

For x = 200,

z = (200 - 200)/50 = 0

Looking at the normal distribution table, the probability corresponding to the z score is 0.5

For x = 275,

z = (275 - 200)/50 = 1.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.9332

Therefore,

P(200 ≤ x ≤ 275) = 0.9332 - 0.5 = 0.4332