Question

4b---Suppose Z is the standardized normal random variable and Z~ N (0, 12). Please use z-table and list detailed steps to get the following probabilities.

1). P(Z < 1.22)=

2). If P(Z < a) = 0.7910, a =

3) If P(Z < b) = 0.3336, b =

4). P(Z > 1.32) = 1-P(Z < 1.32) =

5). P(Z>-1.44) =

6). P(0.35< Z <1.58)=

7). P(-2.3 < Z<-0.88) =

8). P(-2.41 < Z <1.78) =

9). P(Z<1.28) + P(Z> 1.28) =

Answer 1

To solve the given probabilities using the z-table, we need to lookup the corresponding z-scores and calculate the areas under the normal curve. By following the steps outlined in the response, we can find the probabilities for each scenario.

Step-by-step explanation:

1. Using the z-table, we can look up the area to the left of 1.22 as approximately 0.8892.
2. To find a value a such that P(Z < a) = 0.7910, we can use the z-table to find the z-score that corresponds to the area of 0.7910. It is approximately 0.835. So, a ≈ 0.835.
3. To find a value b such that P(Z < b) = 0.3336, we can use the z-table to find the z-score that corresponds to the area of 0.3336. It is approximately -0.433. So, b ≈ -0.433.
4. To find P(Z > 1.32), we can subtract the probability of Z < 1.32 from 1. Using the z-table, we can find that P(Z < 1.32) ≈ 0.9066. So, P(Z > 1.32) ≈ 1 - 0.9066 = 0.0934.
5. To find P(Z > -1.44), we can subtract the probability of Z < -1.44 from 1. Using the z-table, we can find that P(Z < -1.44) ≈ 0.0735. So, P(Z > -1.44) ≈ 1 - 0.0735 = 0.9265.
6. To find the probability of Z being between 0.35 and 1.58, we subtract the probability of Z < 0.35 from the probability of Z < 1.58. Using the z-table, we can find that P(Z < 0.35) ≈ 0.6368 and P(Z < 1.58) ≈ 0.9432. So, P(0.35 < Z < 1.58) ≈ 0.9432 - 0.6368 = 0.3064.
7. To find the probability of Z being between -2.3 and -0.88, we subtract the probability of Z < -2.3 from the probability of Z < -0.88. Using the z-table, we can find that P(Z < -2.3) ≈ 0.0107 and P(Z < -0.88) ≈ 0.1879. So, P(-2.3 < Z < -0.88) ≈ 0.1879 - 0.0107 = 0.1772.
8. To find the probability of Z being between -2.41 and 1.78, we subtract the probability of Z < -2.41 from the probability of Z < 1.78. Using the z-table, we can find that P(Z < -2.41) ≈ 0.0075 and P(Z < 1.78) ≈ 0.9625. So, P(-2.41 < Z < 1.78) ≈ 0.9625 - 0.0075 = 0.9550.
9. To find P(Z < 1.28) + P(Z > 1.28), we can add the probability of Z < 1.28 to 1 minus the probability of Z < 1.28. Using the z-table, we can find that P(Z < 1.28) ≈ 0.8997. So, P(Z < 1.28) + P(Z > 1.28) ≈ 0.8997 + (1 - 0.8997) = 0.8997 + 0.1003 = 1.