Question 1169002: In a recent year, the total scores for a certain standardized test were normally distributed, with a mean of 500 and a standard deviation of 10.3. Answer parts (a)(c) below.
(a) Find the probability that a randomly selected medical student who took the test had a total score that was less than 487.
The probability that a randomly selected medical student who took the test had a total score that was less than is 487
(Round to four decimal places as needed.)
(b) Find the probability that a randomly selected medical student who took the test had a total score that was between 500 and 513.
The probability that a randomly selected medical student who took the test had a total score that was between 500 and 513 is
(Round to four decimal places as needed.)
(c) Find the probability that a randomly selected medical student who took the test had a total score that was more than 53`.
The probability that a randomly selected medical student who took the test had a total score that was more than 531 is
(Round to four decimal places as needed.)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let $X$ be the total score for a randomly selected medical student who took the standardized test. We are given that $X$ is normally distributed with a mean $\mu = 500$ and a standard deviation $\sigma = 10.3$.
**(a) Find the probability that a randomly selected medical student who took the test had a total score that was less than 487.**
We need to find $P(X < 487)$. To do this, we first calculate the z-score for $x = 487$:
$$z = \frac{x - \mu}{\sigma} = \frac{487 - 500}{10.3} = \frac{-13}{10.3} \approx -1.2621$$
Now we need to find the probability $P(Z < -1.2621)$, where $Z$ is a standard normal random variable. Using a standard normal distribution table or a calculator, we find the area to the left of $z = -1.2621$.
$$P(Z < -1.2621) \approx 0.1034$$
The probability that a randomly selected medical student who took the test had a total score that was less than 487 is approximately 0.1034.
**(b) Find the probability that a randomly selected medical student who took the test had a total score that was between 500 and 513.**
We need to find $P(500 < X < 513)$. First, we calculate the z-scores for $x = 500$ and $x = 513$:
For $x = 500$:
$$z_1 = \frac{500 - 500}{10.3} = \frac{0}{10.3} = 0$$
For $x = 513$:
$$z_2 = \frac{513 - 500}{10.3} = \frac{13}{10.3} \approx 1.2621$$
Now we need to find the probability $P(0 < Z < 1.2621) = P(Z < 1.2621) - P(Z < 0)$.
Using a standard normal distribution table or a calculator:
$$P(Z < 1.2621) \approx 0.8966$$
$$P(Z < 0) = 0.5$$
$$P(0 < Z < 1.2621) = 0.8966 - 0.5 = 0.3966$$
The probability that a randomly selected medical student who took the test had a total score that was between 500 and 513 is approximately 0.3966.
**(c) Find the probability that a randomly selected medical student who took the test had a total score that was more than 531.**
We need to find $P(X > 531)$. First, we calculate the z-score for $x = 531$:
$$z = \frac{531 - 500}{10.3} = \frac{31}{10.3} \approx 3.0097$$
Now we need to find the probability $P(Z > 3.0097) = 1 - P(Z < 3.0097)$.
Using a standard normal distribution table or a calculator:
$$P(Z < 3.0097) \approx 0.9987$$
$$P(Z > 3.0097) = 1 - 0.9987 = 0.0013$$
The probability that a randomly selected medical student who took the test had a total score that was more than 531 is approximately 0.0013.
Final Answer: The final answer is:
**(a)** 0.1034
**(b)** 0.3966
**(c)** 0.0013
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