a. To find P(X > 6.5), we first need to standardize the value of 6.5:
z = (x - μ) / σ
= (6.5 - 2.6) / 2.2
= 1.7727
We can then look up this value in the z-table or use a calculator to find the probability. The z-table tells us that the proportion of values to the left of 1.7727 is approximately 0.9625. Therefore, the proportion of values to the right of 1.7727 (or X > 6.5) is 1 - 0.9625 = 0.0375. Rounded to 4 decimal places, P(X > 6.5) is approximately 0.0375.
b. To find P(5.5 ≤ X ≤ 7.5), we need to standardize both values:
z1 = (5.5 - 2.6) / 2.2
= 1.3182
z2 = (7.5 - 2.6) / 2.2
= 2.2273
Using the z-table or a calculator, we can find the proportion of values to the left of z1 and z2:
P(X ≤ 5.5) ≈ 0.9077
P(X ≤ 7.5) ≈ 0.9869
Then, the probability of values falling between 5.5 and 7.5 is:
P(5.5 ≤ X ≤ 7.5) ≈ P(X ≤ 7.5) - P(X ≤ 5.5)
≈ 0.9869 - 0.9077
≈ 0.0792
Rounded to 4 decimal places, P(5.5 ≤ X ≤ 7.5) is approximately 0.0792.
c. To find x such that P(X > x) = 0.0918, we first need to find the z-value for the given probability:
P(Z > z) = 0.0918
Looking up the closest value to 0.0918 in the table, we get 1.31. Therefore, the corresponding z-value is approximately 1.31.
Now we can solve for x:
z = (x - μ) / σ
1.31 = (x - 2.6) / 2.2
Solving for x:
1.31 * 2.2 = x - 2.6
2.882 = x - 2.6
x ≈ 5.482
Rounded to 3 decimal places, x is approximately 5.482.
d. To find x such that P(x ≤ X ≤ 2.6) = 0.3944, we first need to find the corresponding z-values for the given probability:
P(z1 ≤ Z ≤ z2) = 0.3944
Using the z-table or a calculator, we can find the closest values to 0.3944, which are 0.1503 and 0.6331. Therefore, the corresponding z-values are approximately -1.04 and 0.36.
Now we can solve for x:
z1 = (x - μ) / σ
-1.04 = (x - 2.6) / 2.2
Solving for x:
-1.04 * 2.2 = x - 2.6
-2.288 = x - 2.6
x ≈ 0.312
Rounded to 3 decimal places, x is approximately 0.312.
Let X be normally distributed with mean μ = 2.6 and standard deviation σ = 2.2. [You may find it useful to reference the z table.]
a. Find P(X > 6.5). (Round your final answer to 4 decimal places.)
b. Find P(5.5 ≤ X ≤ 7.5). (Round your final answer to 4 decimal places.)
c. Find x such that P(X > x) = 0.0918. (Round your final answer to 3 decimal places.)
d. Find x such that P(x ≤ X ≤ 2.6) = 0.3944. (Negative value should be indicated by a minus sign. Round your final answer to 3 decimal places.)
1 answer
1 answer