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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 170 and 220. Your answer should be a decimal rounded to the fourth decimal place.

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Answer:

0.3811 is the probability that a randomly selected applicant will have a rating between 170 and 220.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 200

Standard Deviation, σ = 50

We are given that the distribution of ratings is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

P(rating is between 170 and 220)

[tex]P(170 \leq x \leq 220)\\\\ = P(\displaystyle\frac{170 - 200}{50} \leq z \leq \displaystyle\frac{220-200}{50}) \\\\= P(-0.6 \leq z \leq 0.4)\\\\= P(z \leq 0.4) - P(z < -0.6)\\= 0.6554 - 0.2743 = 0.3811 = 38.11\%[/tex]

0.3811 is the probability that a randomly selected applicant will have a rating between 170 and 220.