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.)
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.
To solve these questions, we will need to use the standard normal distribution.
a. To find P(X > 6.5), we need to find the area to the right of the value 6.5. Since we have the mean (μ) and standard deviation (σ), we need to standardize the value 6.5 using the z-score formula:
z = (x - μ) / σ
z = (6.5 - 2.6) / 2.2
= 1.7727
Now, referring to the z-table, we can find the area to the left of the z-score 1.7727 (since we are looking for the area to the right):
P(X > 6.5) = 1 - P(Z ≤ 1.7727)
= 1 - 0.9625
= 0.0375
Therefore, P(X > 6.5) ≈ 0.0375.
b. To find P(5.5 ≤ X ≤ 7.5), we need to find the area between these two values. Again, we standardize the values using the z-score formula:
z1 = (5.5 - 2.6) / 2.2
= 1.3182
z2 = (7.5 - 2.6) / 2.2
= 2.2273
Next, we find the areas to the left of these z-scores:
P(5.5 ≤ X ≤ 7.5) = P(Z ≤ 2.2273) - P(Z ≤ 1.3182)
= 0.9871 - 0.9088
= 0.0783
Therefore, P(5.5 ≤ X ≤ 7.5) ≈ 0.0783.
c. To find x such that P(X > x) = 0.0918, we need to find the corresponding z-score using the z-table. We want to find the z-score with an area of 0.0918 to the left.
Looking up the closest value to 0.0918 in the z-table, we find that it corresponds to a z-score of -1.34. Now we can solve for x using the z-score formula:
-1.34 = (x - 2.6) / 2.2
Solving for x, we get:
-1.34 * 2.2 = x - 2.6
-2.948 = x - 2.6
x = -2.948 + 2.6
x ≈ -0.348
Therefore, x ≈ -0.348.
d. To find x such that P(x ≤ X ≤ 2.6) = 0.3944, we need to find the corresponding z-scores for the given probabilities.
First, using the z-table, we find the closest value to 0.3944, which corresponds to a z-score of 0.25. We can now solve for x using the z-score formula:
0.25 = (x - 2.6) / 2.2
Solving for x, we get:
0.25 * 2.2 = x - 2.6
0.55 = x - 2.6
x = 0.55 + 2.6
x ≈ 3.15
Therefore, x ≈ 3.15.
To solve these problems, we can use the standard normal distribution, also known as the z-distribution. The z-distribution has a mean of 0 and a standard deviation of 1. We can convert our given normal distribution to the standard normal distribution by using the z-score formula:
z = (X - μ) / σ
where X is the given value, μ is the mean, and σ is the standard deviation.
a. To find P(X > 6.5), we need to find the area under the right tail of the standard normal distribution. We can standardize the value 6.5 using the z-score formula:
z = (6.5 - 2.6) / 2.2 = 1.7727
Looking up the z-score 1.7727 on the z-table, we find that the area to the left of this value is 0.9625. Therefore, the area to the right of this z-score is:
1 - 0.9625 = 0.0375
So, P(X > 6.5) = 0.0375.
b. To find P(5.5 ≤ X ≤ 7.5), we need to find the area between these two values on the standard normal distribution. We can standardize these values using the z-score formula:
z1 = (5.5 - 2.6) / 2.2 = 1.3182
z2 = (7.5 - 2.6) / 2.2 = 2.1364
Looking up the z-score 1.3182 on the z-table, we find that the area to the left of this value is 0.9099. Looking up the z-score 2.1364, we find that the area to the left of this value is 0.9846. Therefore, the area between these two z-scores is:
0.9846 - 0.9099 = 0.0747
So, P(5.5 ≤ X ≤ 7.5) = 0.0747.
c. To find x such that P(X > x) = 0.0918, we can use the inverse z-score formula. We need to find the z-score corresponding to an area of 0.0918 in the right tail of the standard normal distribution.
Looking up the z-score corresponding to an area of 0.0918 on the z-table, we find that the z-score is approximately -1.343.
Using the z-score formula:
-1.343 = (x - 2.6) / 2.2
Solving for x:
x = -1.343 * 2.2 + 2.6
x ≈ 2.288
So, x ≈ 2.288.
d. To find x such that P(x ≤ X ≤ 2.6) = 0.3944, we need to use the inverse z-score formula. We want to find the z-score corresponding to an area of 0.3944 in the left tail of the standard normal distribution.
Looking up the z-score corresponding to an area of 0.3944 on the z-table, we find that the z-score is approximately -0.281.
Using the z-score formula:
-0.281 = (x - 2.6) / 2.2
Solving for x:
x = -0.281 * 2.2 + 2.6
x ≈ 2.062
So, x ≈ 2.062, which indicates a negative value in this case.