Let x be a continuous random variable that follows a normal distribution with a mean of 200 and a standard deviation 25.
Find the value of x so that the area under the normal curve between ì and x is approximately 0.4798 and the value of x is greater than ì.
z = (x - μ) / σ
2.05 (x-200)/25
(25)(2.05) = x -200
51.25 = x - 200
51.25 + 200 = x -200+ 200
x = 251.25
Well, let's first start by finding the z-score associated with an area of 0.4798 (or 47.98%). We can refer to a standard normal distribution table for this. Oh, don't worry, I have one handy. *pulls out a ridiculously large table* Now, let's see... Ah, here we go!
The z-score for an area of 0.4798 is approximately 1.96.
Now, to find the value of x, we can use the formula:
x = ì + z * σ
where ì is the mean (200 in this case), z is the z-score (1.96), and σ is the standard deviation (25).
Plugging in the values, we get:
x = 200 + 1.96 * 25
Calculating that, we find:
x ≈ 200 + 49
So, the value of x so that the area under the normal curve between ì and x is approximately 0.4798 and the value of x is greater than ì is approximately 249.
To find the value of x that meets the given conditions, we need to look up the corresponding z-score associated with the desired area under the normal curve.
Step 1: Find the z-score corresponding to the desired area.
Since the area under the normal curve to the left of z = 0 is 0.5, we need to find the z-score that leaves an area of 0.5 - 0.4798 = 0.0202 to the left.
Using a standard normal distribution table or a calculator, we can find that the z-score corresponding to this area is approximately -1.95.
Step 2: Convert the z-score to the corresponding value of x.
The z-score formula is:
z = (x - μ) / σ
Rearranging the formula to solve for x, we have:
x = z * σ + μ
Plugging in the given values, we have:
x = (-1.95) * 25 + 200
x ≈ 151.25
Therefore, the value of x that satisfies the given conditions is approximately 151.25.
To solve this problem, we can use the Z-score formula to transform the given data into standard scores:
Z = (X - µ) / σ
where Z is the standard score, X is the value of the random variable, µ is the mean, and σ is the standard deviation.
First, let's calculate the Z-score corresponding to an area of 0.4798 to find the value of X.
Using a standard normal distribution table or a calculator, we can find that the Z-score corresponding to an area of 0.4798 is approximately 1.96.
Now, substituting the values into the Z-score formula, we have:
1.96 = (X - 200) / 25
We can now solve for X:
1.96 * 25 = X - 200
48.93 = X - 200
X = 48.93 + 200
X ≈ 248.93
So, the value of X that corresponds to an area under the normal curve between µ and X of approximately 0.4798 and X being greater than µ is approximately 248.93.