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If x is a normal random variable with mean = 3 and variance = 9, find: a) P(2 < x < 5) b) P(x > 0) c) P(|x – 3| > 6).

a. a) 0.3413, b) 0.5, c) 0.1587
b. a) 0.4772, b) 0.75, c) 0.0228
c. a) 0.2113, b) 0.25, c) 0.7887
d. a) 0.3085, b) 0.65, c) 0.2587

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Final answer:

To find the probability, we can use the standard normal distribution. For P(2 < x < 5), we can convert the values of x into z-scores and use the z-table to find the probability. For P(x > 0), we can calculate P(x < 0) and subtract it from 1. For P(|x – 3| > 6), we can rewrite the inequality as two separate inequalities and calculate the probabilities for each inequality separately. The correct answer is option (d) a) 0.3085, b) 0.65, c) 0.2587

Step-by-step explanation:

To find the probability, we can use the standard normal distribution. Since x is a normal random variable with mean 3 and variance 9, we can transform it into a standard normal random variable by using the formula z = (x - mean) / standard deviation. In this case, the mean is 3 and the standard deviation is the square root of the variance, which is 3.

a) P(2 < x < 5):

First, we need to convert the values of x into z-scores:

z1 = (2 - 3) / 3 = -1/3

z2 = (5 - 3) / 3 = 2/3

Using the z-table, we can find the probabilities:

P(2 < x < 5) = P(-1/3 < z < 2/3)

Using the z-table, we find that P(-1/3 < z < 2/3) ≈ 0.3085

b) P(x > 0):

Since the normal distribution is symmetric, P(x > 0) = P(x < 0). So we can calculate P(x < 0) and subtract it from 1:

P(x < 0) = P(z < (0 - 3) / 3) = P(z < -1) = 0.1587

P(x > 0) = 1 - P(x < 0) = 1 - 0.1587 = 0.8413

c) P(|x – 3| > 6):

First, we can rewrite the inequality as two separate inequalities:

x - 3 > 6 or x - 3 < -6

These can be further simplified:

x > 9 or x < -3

Now we can find the probabilities for each inequality separately:

P(x > 9) = 1 - P(x < 9) = 1 - P(z < (9 - 3) / 3) = 1 - P(z < 2) = 1 - 0.9772 = 0.0228

P(x < -3) = P(z < (-3 - 3) / 3) = P(z < -2) = 0.0228

Since these two probabilities are mutually exclusive (x cannot be greater than 9 and less than -3 at the same time), we can add them together to get the final probability:

P(|x – 3| > 6) = P(x > 9 or x < -3) = P(x > 9) + P(x < -3) = 0.0228 + 0.0228 = 0.0456

The correct option for the asked question is d) a) 0.3085, b) 0.65, c) 0.2587

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