To solve this problem, we can use the z-score formula, which is given as:
z = (X - μ) / σ
where X is the random variable, μ is the mean, and σ is the standard deviation.
a. To find P(X ≤ 0), we need to find the z-score corresponding to 0 and then look up the corresponding probability in the z-table.
z = (0 - 26) / 13
z = -26 / 13
z = -2
Looking up the z-score of -2 in the z-table, we find that the corresponding probability is 0.0228.
Therefore, P(X ≤ 0) = 0.0228.
b. To find P(X > 13), we can find the probability of X ≤ 13 and subtract it from 1.
z = (13 - 26) / 13
z = -13 / 13
z = -1
Looking up the z-score of -1 in the z-table, we find that the corresponding probability is 0.1587.
P(X ≤ 13) = 0.1587
P(X > 13) = 1 - P(X ≤ 13) = 1 - 0.1587 = 0.8413
Therefore, P(X > 13) = 0.8413.
c. To find P(13 ≤ X ≤ 26), we can find the probability of X ≤ 26 and subtract the probability of X ≤ 13.
P(X ≤ 13) = 0.1587 (from previous calculations)
P(X ≤ 26) = 0.5 (since it is half of the distribution)
P(13 ≤ X ≤ 26) = P(X ≤ 26) - P(X ≤ 13) = 0.5 - 0.1587 = 0.3413
Therefore, P(13 ≤ X ≤ 26) = 0.3413.
d. To find P(26 ≤ X ≤ 39), we can find the probability of X ≤ 39 and subtract the probability of X ≤ 26.
P(X ≤ 26) = 0.5 (since it is half of the distribution)
P(X ≤ 39) = 0.8413 (from previous calculations)
P(26 ≤ X ≤ 39) = P(X ≤ 39) - P(X ≤ 26) = 0.8413 - 0.5 = 0.3413
Therefore, P(26 ≤ X ≤ 39) = 0.3413.
Let X be normally distributed with mean μ = 26 and standard deviation σ = 13. [You may find it useful to reference the z table.]
a. Find P(X ≤ 0). (Round your final answer to 4 decimal places.)
b. Find P(X > 13). (Round your final answer to 4 decimal places.)
c. Find P(13 ≤ X ≤ 26). (Round your final answer to 4 decimal places.)
d. Find P(26 ≤ X ≤ 39). (Round your final answer to 4 decimal places.)
1 answer
1 answer