Final answer:
To find the probability using the z-table, we need to calculate the z-scores for each question and then look up the corresponding area under the curve in the z-table. P(Z < -1.35) = 0.0885, P(Z > 1.38) = 0.0843, P(-0.4< Z< 1.5) = 0.5886, and P(|Z| > 2) = 0.0456.
Step-by-step explanation:
To find the probability using the z-table, we need to calculate the z-scores for each question and then look up the corresponding area under the curve in the z-table.
a) P(Z < -1.35):
The z-score for -1.35 can be found using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
Substituting the values, we get z = (-1.35 - 0) / 1 = -1.35.
Looking up the z-score in the z-table, we find the area to the left of -1.35 is 0.0885.
Therefore, P(Z < -1.35) = 0.0885.
b) P(Z > 1.38):
Using the same formula, we find the z-score for 1.38 is (1.38 - 0) / 1 = 1.38.
Looking up the z-score in the z-table, we find the area to the left of 1.38 is 0.9157.
To find the area to the right of 1.38, we subtract the area to the left from 1: 1 - 0.9157 = 0.0843.
Therefore, P(Z > 1.38) = 0.0843.
c) P(-0.4< Z< 1.5):
Using the same formula, we find the z-score for -0.4 is (-0.4 - 0) / 1 = -0.4.
Looking up the z-score in the z-table, we find the area to the left of -0.4 is 0.3446.
Using the same formula, we find the z-score for 1.5 is (1.5 - 0) / 1 = 1.5.
Looking up the z-score in the z-table, we find the area to the left of 1.5 is 0.9332.
To find the area between these two z-scores, we subtract the area to the left of -0.4 from the area to the left of 1.5: 0.9332 - 0.3446 = 0.5886.
Therefore, P(-0.4< Z< 1.5) = 0.5886.
d) P(|Z| > 2):
The absolute value bars indicate that we're looking for the area in the tails of the curve.
To find the area in the right tail, we subtract the area to the left of 2 from 1: 1 - 0.9772 = 0.0228.
To find the area in the left tail, we subtract the area to the right of -2 from 1: 1 - 0.9772 = 0.0228.
Then, to find the area in both tails, we add the two areas together: 0.0228 + 0.0228 = 0.0456.
Therefore, P(|Z| > 2) = 0.0456.