Answer to Question #162652 in Statistics and Probability for Sunny

Question

Assuming GMAT scores are normally distributed having mean of 540 and standard deviation of 100, find the probability of obtaining

a. A score of 750 or higher?

b. A score of 590 or less?

c. A score of 540 or less?

Answer 1

Let $\varPhi$ be the CDF of the standard normal distribution. Let $\xi$ be a random variable such that $mean \lparen \xi \rparen = 540$ and $std \lparen \xi \rparen = 100$.

a) The probability of obtaining a score of 750 or less is equal to $P \lparen \xi \geq 750 \rparen = \lbrace$ the compliment rule $\rbrace = 1 - P \lparen \xi \lt 750 \rparen = \\ = 1 - P \lparen \frac{\xi - mean \lparen \xi \rparen}{std \lparen \xi \rparen} \lt \frac{750 - mean \lparen \xi \rparen}{std \lparen \xi \rparen} \rparen = \\ = 1 - P \lparen \frac{\xi - 540}{100} \lt \frac{750 - 540}{100} \rparen = \\ = 1 - P \lparen \frac{\xi - 540}{100} \lt 2.1\rparen = \\ =1 - \varPhi \lparen 2.1 \rparen \approx \\ \approx 1 - 0.9821 = 0.0179$

b) The probability of obtaining a score of 590 or less is equal to $P \lparen \xi \leq 590 \rparen = \\ = P \lparen \xi \lt 590 \rparen + P \lparen \xi = 590 \rparen = \lbrace$ the normal distribution is a type of continuous probability distribution, so $P \lparen \xi = 590 \rparen = 0 \rbrace = \\ = P \lparen \xi \lt 590 \rparen = \\ = P \lparen \frac{\xi - 540}{100} < 0.5 \rparen = \\ = \varPhi \lparen 0.5 \rparen \approx 0.6915$

c) The probability of obtaining a score of 540 or less is equal to $P \lparen \xi \leq 540 \rparen = \\ = P \lparen \xi \lt 540 \rparen = \\ = P \lparen \frac{\xi - 540}{100} < 0 \rparen = \\ = \varPhi \lparen 0 \rparen = 0.5$