a. To find P(X ≤ 100), we need to calculate the z-score first. The z-score formula is given by:
z = (x - μ) / σ
Plugging in the values, we have:
z = (100 - 102) / 34 = -0.0588
Now, we can use the z-table to find the probability. Looking up the z-score of -0.0588, we find that the probability is 0.5279.
Therefore, P(X ≤ 100) = 0.5279.
b. To find P(95 ≤ X ≤ 110), we need to calculate the z-scores for both values:
For 95, z = (95 - 102) / 34 = -0.2059
For 110, z = (110 - 102) / 34 = 0.2353
Using the z-table, we can find the probabilities corresponding to these z-scores:
P(X ≤ 95) = 0.4207
P(X ≤ 110) = 0.5899
To find P(95 ≤ X ≤ 110), we subtract the two probabilities:
P(95 ≤ X ≤ 110) = P(X ≤ 110) - P(X ≤ 95) = 0.5899 - 0.4207 = 0.1692
Therefore, P(95 ≤ X ≤ 110) = 0.1692.
c. We are given that P(X ≤ x) = 0.360. To find x, we need to find the z-score corresponding to this probability.
Using the z-table, we find that the z-score for a cumulative probability of 0.360 is approximately -0.3300.
Now, we can use the z-score formula to find x:
-0.3300 = (x - 102) / 34
Solving for x, we have:
-0.3300 * 34 = x - 102
-11.2200 = x - 102
x = -11.2200 + 102
x = 90.7800
Therefore, x ≈ 90.780.
d. We are given that P(X > x) = 0.830. To find x, we need to find the z-score corresponding to this probability.
Since we are looking for the probability greater than x, we subtract the given probability from 1:
P(X > x) = 1 - P(X ≤ x) = 1 - 0.830 = 0.170
Using the z-table, we can find the z-score corresponding to this probability:
The z-score for a cumulative probability of 0.170 is approximately -0.9382.
Now, we can use the z-score formula to find x:
-0.9382 = (x - 102) / 34
Solving for x, we have:
-0.9382 * 34 = x - 102
-31.893 = x - 102
x = -31.893 + 102
x = 70.107
Therefore, x ≈ 70.107.
Let X be normally distributed with mean μ = 102 and standard deviation σ = 34. [You may find it useful to reference the z table.]
a. Find P(X ≤ 100). (Round your final answer to 4 decimal places.)
b. Find P(95 ≤ X ≤ 110). (Round your final answer to 4 decimal places.)
c. Find x such that P(X ≤ x) = 0.360. (Round your final answer to 3 decimal places.)
d. Find x such that P(X > x) = 0.830. (Round your final answer to 3 decimal places.)
1 answer
1 answer