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Given that is a standard normal random variable, compute the following probabilities (to 4 decimals). a. P(25 -1.0) b. P(Z > -1.0) c. P(z2 -1.5) d. P(x2 -2.5) e. P(-3<230) Navigation Menu

User AreYouSure
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Final answer:

To compute the probabilities for the given standard normal random variable, we can use the cumulative distribution function (CDF) of the standard normal distribution. We can find probabilities such as P(Z > -1.0), P(Z > 1.5), and P(-3 < Z < 2.0) by either subtracting the CDF from 1 or by finding the difference between two CDFs. Using a standard normal probability table or a calculator, we can compute these probabilities.

Step-by-step explanation:

To compute the probabilities for the given standard normal random variable, we can use the cumulative distribution function (CDF) of the standard normal distribution. The CDF gives the probability that a standard normal random variable is less than or equal to a given value.

a. To find P(Z > -1.0), we can subtract the CDF at -1.0 from 1: P(Z > -1.0) = 1 - P(Z <= -1.0). Using a standard normal probability table or a calculator, we find P(Z <= -1.0) = 0.1587. Therefore, P(Z > -1.0) = 1 - 0.1587 = 0.8413.

b. To find P(Z > 1.5), we can use the same approach: P(Z > 1.5) = 1 - P(Z <= 1.5). Using the table or calculator, we find P(Z <= 1.5) = 0.9332. Therefore, P(Z > 1.5) = 1 - 0.9332 = 0.0668.

c. To find P(-3 < Z < 2.0), we need to find the probabilities for both -3 and 2.0 separately and then subtract them: P(-3 < Z < 2.0) = P(Z < 2.0) - P(Z < -3). Using the table or calculator, we find P(Z < 2.0) = 0.9772 and P(Z < -3) = 0.0013. Therefore, P(-3 < Z < 2.0) = 0.9772 - 0.0013 = 0.9759.

User Sean McCleary
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Final answer:

To compute the given probabilities, use the standard normal distribution table or calculator. The probabilities are: a. 0.1525, b. 0.8413, c. 0.9104, d. 0.9189, e. 0.9893.

Step-by-step explanation:

Since Z is a standard normal random variable, the probability P(Z> -1.0) indicates that we should locate the area under the standard normal curve that is to the right of -1.0. To compute the given probabilities, we need to use the standard normal distribution table or calculator:

a. P(2.5 < Z < -1.0) = P(Z > -1.0) - P(Z > 2.5) = 0.1587 - 0.0062 = 0.1525

b. P(Z > -1.0) = 0.8413

c. P(-1.5 < Z < 2) = P(Z > -1.5) - P(Z > 2) = 0.9332 - 0.0228 = 0.9104

d. P(-2.5 < Z < 2) = P(Z > -2.5) - P(Z > 2) = 1 - 0.0811 = 0.9189

e. P(-3 < Z < 2.30) = P(Z > -3) - P(Z > 2.30) = 0.9987 - 0.0094 = 0.9893

User Miles
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