Find the following z values for the standard normal variable Z. (You may find it useful to reference the z table. Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)
P(Z<=z)=0.8914
P(Z>z)=0.8001
P(-z<=Z<=z)=0.94
P(0<=Z<=z)=0.4067
To find the z value for P(Z<=z) = 0.8914, you would look up the value 0.8914 in the standard normal distribution table. The closest value to this probability is 0.8915, which corresponds to a z value of approximately 1.25.
To find the z value for P(Z>z) = 0.8001, you would first find the value for P(Z<=z) by subtracting this probability from 1. So P(Z<=z) = 1 - 0.8001 = 0.1999. Looking up this value in the table, the closest value is 0.1998, which corresponds to a z value of approximately -0.90. Therefore, the z value for P(Z>z) = 0.8001 is approximately -0.90.
To find the z values for P(-z<=Z<=z) = 0.94, you would find the corresponding area for the tail probabilities at both ends and then subtract them from 0.94. The tail probability is (1 - 0.94)/2 = 0.03, so each end would have a tail probability of 0.03. Looking up the tail probability of 0.03 in the table, the closest value is 0.0294, which corresponds to a z value of approximately 1.88 (positive end) and -1.88 (negative end). Therefore, the z values for P(-z<=Z<=z) = 0.94 are approximately -1.88 and 1.88.
To find the z value for P(0<=Z<=z) = 0.4067, you would subtract the probability at the lower end (0) from the given probability. So, P(Z<=z) = 0.4067 - 0 = 0.4067. Looking up this value in the table, the closest value is 0.4082, which corresponds to a z value of approximately 1.20. Therefore, the z value for P(0<=Z<=z) = 0.4067 is approximately 1.20.
To find the z-values for the given probabilities, we can use the z-table or a statistical calculator. Here are the steps for each scenario:
1. P(Z <= z) = 0.8914:
From the z-table, find the closest value to 0.8914. The closest value is 0.8915, which corresponds to a z-value of 1.23 (positive).
2. P(Z > z) = 0.8001:
Subtract the probability from 1:
1 - 0.8001 = 0.1999
From the z-table, find the closest value to 0.1999. The closest value is 0.1997, which corresponds to a z-value of -0.84 (negative).
3. P(-z <= Z <= z) = 0.94:
Subtract the probabilities at both ends from 1:
1 - 0.94 = 0.06
Since the probability is symmetrical, we can find the value corresponding to 0.03 (half of 0.06) in the z-table. The closest value is 0.9893, which corresponds to a z-value of 1.81 (positive). This means that the negative value of z is also -1.81.
4. P(0 <= Z <= z) = 0.4067:
Since the probability starts from 0, we can directly find the value corresponding to 0.4067 in the z-table. The closest value is 0.6554, which corresponds to a z-value of 0.25 (positive).
So, the z-values for the given probabilities are:
1. P(Z <= z) = 0.8914 → z = 1.23
2. P(Z > z) = 0.8001 → z = -0.84
3. P(-z <= Z <= z) = 0.94 → z = -1.81 and z = 1.81
4. P(0 <= Z <= z) = 0.4067 → z = 0.25 (positive)
To find the z values for the given probabilities, we can use the standard normal distribution table (also known as the z-table). The z-table provides the area under the curve (probability) for different values of z.
1. P(Z <= z) = 0.8914:
Go to the z-table and find the closest value to 0.8914. The nearest value is 0.8915, located in the leftmost column of the table. Now, move horizontally to the right until you find the corresponding value in the row labeled with 0.09. The z-value for a cumulative probability of 0.8914 is approximately 1.22.
2. P(Z > z) = 0.8001:
Since the given probability is for the probability that Z is greater than z, we need to find the complement of this probability. Complementing the probability means subtracting it from 1.
1 - P(Z > z) = 1 - 0.8001 = 0.1999
Now, go to the z-table and search for the value closest to 0.1999. The closest value is 0.2005, located in the leftmost column of the table. Move horizontally until you find the corresponding value in the row labeled with 0.00. The z-value for a cumulative probability of 0.1999 is approximately -0.84.
Therefore, z is approximately -0.84.
3. P(-z <= Z <= z) = 0.94:
For this probability, we need to find the z-values for the area between the negative z-value and positive z-value. First, subtract the cumulative probability from 1 to find the complement.
1 - P(-z <= Z <= z) = 1 - 0.94 = 0.06
Divide this complement in half to get the probability in each tail: 0.06 / 2 = 0.03
Now, go to the z-table and find the value closest to 0.03. The nearest value is 0.0294, located in the leftmost column of the table. Move horizontally to the right until you find the corresponding value in the row labeled with 0.00. The z-value for a cumulative probability of 0.03 is approximately -1.88.
Therefore, the negative z-value is approximately -1.88. Since the standard normal distribution is symmetrical, the positive z-value is the same but with a positive sign. Thus, the positive z-value is approximately 1.88.
4. P(0 <= Z <= z) = 0.4067:
To find the z-value for this probability, subtract the cumulative probability from 1 to find the complement.
1 - P(0 <= Z <= z) = 1 - 0.4067 = 0.5933
Now, go to the z-table and find the value closest to 0.5933. The nearest value is 0.5931, located in the leftmost column of the table. Move horizontally until you find the corresponding value in the row labeled with 0.04. The z-value for a cumulative probability of 0.5933 is approximately 0.23.
Therefore, z is approximately 0.23.