Question

Calculate the following:

a) If Z is distributed normally with mean 0 and variance 1, find:
i) P(Z < 1.45)
ii) P(Z > -2.65)

b) If X is distributed normally with mean 4 and variance 16, find:
i) P(X > 7)
ii) P(-13.2 < X)

Answer 1

The probabilities for the normal distribution questions can be found by converting the given values to Z-scores and then using the standard normal distribution table or computational software. For a standard normal distribution with mean 0 and variance 1, P(Z < 1.45) and P(Z > -2.65) can be obtained, as well as P(X > 7) and P(-13.2 < X) for a distribution with mean 4 and variance 16.

Step-by-step explanation:

To calculate the probabilities for a normally distributed variable Z with a mean of 0 and variance of 1 (standard deviation = 1):

1. P(Z < 1.45): This is the probability that the standard normal variable Z is less than 1.45. It can be found using the standard normal distribution table or software that calculates normal probabilities.

2. P(Z > -2.65): Similarly, this is the probability that Z is greater than -2.65. Again, standard normal distribution tables or software can provide this probability.

For a normally distributed variable X with a mean of 4 and variance of 16 (standard deviation = 4):

1. P(X > 7): To find this, we first convert the value of X to a Z-score using the formula: Z = (X - mean) / standard deviation, which in this case is Z = (7 - 4) / 4 = 0.75. The probability P(X > 7) is the same as P(Z > 0.75).

2. P(-13.2 < X): To express as a Z-score, Z = (-13.2 - 4) / 4 = -4.3. This probability is P(Z > -4.3), which is essentially 1 since -4.3 is far into the left tail of the standard normal distribution.