Given that z is a standard normal random variable, compute the following probabilities (to 4 decimals).
P(z -1.0)
P(z -1.0)
P(z -1.5)
P(z -2.5)
P(-3 < z 0)
To find the probabilities for a standard normal random variable, we use the Z-table (also known as the standard normal table) which provides the area (probability) to the left of a given Z-score.
Here's the link to a Z-table for your reference: http://www.z-table.com/
Now, we will use the Z-table to find the probabilities for each given Z-score.
1. P(z < -1.0): From the Z-table, the probability is 0.1587.
2. P(z > -1.0): Since the area under the entire curve sums up to 1, P(z > -1.0) = 1 - P(z < -1.0) = 1 - 0.1587 = 0.8413.
3. P(z > -1.5): From the Z-table, P(z < -1.5) = 0.0668. Therefore, P(z > -1.5) = 1 - 0.0668 = 0.9332.
4. P(z > -2.5): From the Z-table, P(z < -2.5) = 0.0062. Therefore, P(z > -2.5) = 1 - 0.0062 = 0.9938.
5. P(-3 < z < 0): Here, we need to find the probability between two Z-scores. P(-3 < z < 0) = P(z < 0) - P(z < -3). From the Z-table, P(z < 0) = 0.5000 and P(z < -3) = 0.0013. Therefore, P(-3 < z < 0) = 0.5000 - 0.0013 = 0.4987.
So, the requested probabilities are:
P(z < -1.0) = 0.1587
P(z > -1.0) = 0.8413
P(z > -1.5) = 0.9332
P(z > -2.5) = 0.9938
P(-3 < z < 0) = 0.4987
To compute probabilities for a standard normal random variable, you can use a standard normal distribution table or a statistical software.
For P(z < -1.0):
1. Look up the value of -1.0 in the standard normal distribution table.
2. The table will give you the corresponding cumulative probability.
3. Round the probability to 4 decimal places.
For P(z > -1.0), subtract the cumulative probability from 1.
For P(z < -1.5) and P(z < -2.5), repeat steps 1-3 using -1.5 and -2.5, respectively.
For P(-3 < z < 0):
1. Look up the value of -3 and 0 in the standard normal distribution table.
2. Find the corresponding cumulative probabilities for both values.
3. Subtract the cumulative probability for -3 from the cumulative probability for 0.
4. Round the probability to 4 decimal places.