(4 pts) The score on an exam from a certain MAT 112 class, X, is normally distributed with \mu = 77.6 and \sigma = 10.9.

NOTE: Assume for the sake of this problem that the score is a continuous variable. A score can thus take on any value on the continuum. (In real life, scores are often treated as if they were continuous values but are actually discrete in most cases.)
(a) Write the event ''a score less than 69.6'' in terms of X: .

(b) Find the probability of this event:

(c) Find the probability that a randomly chosen score is greater than 86.1: .

(d) Find the probability that a randomly chosen score is between 69.6 and 86.1:

(a) The event ''a score less than 69.6'' can be written in terms of X as X < 69.6.

(b) To find the probability of this event, we need to calculate the cumulative probability up to the value of 69.6 using the standard normal distribution.

P(X < 69.6) = P(Z < (69.6 - 77.6) / 10.9) = P(Z < -0.73)

Using a standard normal distribution table or a calculator, we find that the probability P(Z < -0.73) is approximately 0.2327.

Therefore, the probability of the event ''a score less than 69.6'' is approximately 0.2327.

(c) To find the probability that a randomly chosen score is greater than 86.1, we need to calculate the cumulative probability beyond the value of 86.1 using the standard normal distribution.

P(X > 86.1) = 1 - P(X < 86.1) = 1 - P(Z < (86.1 - 77.6) / 10.9) = 1 - P(Z < 0.78)

Using a standard normal distribution table or a calculator, we find that the probability P(Z < 0.78) is approximately 0.7814.

Therefore, the probability of the event ''a randomly chosen score is greater than 86.1'' is approximately 0.2186.

(d) To find the probability that a randomly chosen score is between 69.6 and 86.1, we need to calculate the cumulative probability between these two values using the standard normal distribution.

P(69.6 < X < 86.1) = P(Z < (86.1 - 77.6) / 10.9) - P(Z < (69.6 - 77.6) / 10.9) = P(Z < 0.78) - P(Z < -0.73)

Using a standard normal distribution table or a calculator, we find that P(Z < 0.78) is approximately 0.7814 and P(Z < -0.73) is approximately 0.2327.

Therefore, the probability of the event ''a randomly chosen score is between 69.6 and 86.1'' is approximately 0.7814 - 0.2327 = 0.5487.

(a) To write the event "a score less than 69.6" in terms of X, we can use the inequality notation. Let's represent the event as X < 69.6.

(b) To find the probability of the event X < 69.6, we need to find the area under the normal curve to the left of 69.6. This area represents the probability of obtaining a score less than 69.6. We can use the cumulative distribution function (CDF) of the normal distribution.

To calculate this probability, we need to standardize the score using the z-score formula:
z = (X - μ) / σ

Substituting the given values, we get:
z = (69.6 - 77.6) / 10.9

Calculating the z-score, we get:
z = -0.733

Using the CDF table or a calculator, we find that the probability corresponding to a z-score of -0.733 is approximately 0.2317.

Therefore, the probability of a score less than 69.6 is 0.2317.

(c) To find the probability that a randomly chosen score is greater than 86.1, we need to find the area under the normal curve to the right of 86.1. We can again use the CDF of the normal distribution.

First, we calculate the z-score using the formula:
z = (X - μ) / σ

Substituting the given values, we get:
z = (86.1 - 77.6) / 10.9

Calculating the z-score, we get:
z = 0.779

Using the CDF table or a calculator, we find that the probability corresponding to a z-score of 0.779 is approximately 0.7833.

Therefore, the probability of a randomly chosen score being greater than 86.1 is 0.7833.

(d) To find the probability that a randomly chosen score is between 69.6 and 86.1, we need to find the area under the normal curve between these two values. This can be done by finding the difference between the cumulative probabilities at each value.

Using the CDF of the normal distribution, we find the probability of a score less than 69.6 as 0.2317, and the probability of a score less than 86.1 as 0.7833.

Therefore, the probability of a randomly chosen score being between 69.6 and 86.1 can be calculated as:
P(69.6 < X < 86.1) = P(X < 86.1) - P(X < 69.6)
= 0.7833 - 0.2317
= 0.5516

Therefore, the probability of a randomly chosen score being between 69.6 and 86.1 is 0.5516.

Use same process as in previous post.