The time it takes to install the new telephone is found to be normally distributed with the mean of x minutes and a standard deviation of 8 minutes. If the probability of installing a new telephone more than 65 minutes is 0.0062, find the value of x.
To solve this problem, we need to find the z-score corresponding to the probability of installing a new telephone more than 65 minutes.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
In this case, x = 65, σ = 8, and we want to find the value of μ.
To find the z-score, we rearrange the formula:
z = (x - μ) / σ
z = (65 - μ) / 8
Next, we look up the z-score in the standard normal distribution table to find the corresponding probability.
Using the table, we find that the z-score corresponding to a probability of 0.0062 is approximately -2.72.
Therefore, we can set up the equation:
-2.72 = (65 - μ) / 8
Now, we can solve for μ:
-2.72 * 8 = 65 - μ
-21.76 = 65 - μ
μ = 65 - (-21.76)
μ = 65 + 21.76
μ = 86.76
Therefore, the value of x (the mean time it takes to install a new telephone) is 86.76 minutes.
To find the value of x, we need to use the z-score formula and the standard normal distribution.
The z-score formula is given by:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the given value
- μ is the mean
- σ is the standard deviation
We want to find the value of x when the probability is 0.0062, which corresponds to a z-score of -2.57 (since the probability is on the right side of the distribution).
Using the z-score formula, we can rearrange it to solve for x:
x = z * σ + μ
Substituting the given values:
- z = -2.57
- σ = 8
To find μ, we need to subtract -2.57 * 8 from x:
x = -2.57 * 8 + μ
Simplifying the equation:
x = -20.56 + μ
Finally, we know that the probability of installing a new telephone more than 65 minutes is 0.0062. Since 65 minutes corresponds to a z-score of -2.57, we can conclude that this z-score is equal to -2.57.
So, to find the value of x, we set x equal to 65 and solve for μ:
65 = -20.56 + μ
Adding 20.56 to both sides:
85.56 = μ
Therefore, the value of x (the mean) is approximately 85.56 minutes.
To solve this problem, we need to find the value of x.
Given:
Mean (μ) = x minutes
Standard Deviation (σ) = 8 minutes
Probability (P) for installing a new telephone > 65 minutes = 0.0062
We will use the Z-score formula to find the corresponding Z-score for the given probability:
Z = (X - μ) / σ
Here, X is the unknown value (installing time), μ is the mean, and σ is the standard deviation.
Now, we need to find the Z-score such that the area under the normal distribution curve to the right of the Z-score is 0.0062.
From the Z-table or a calculator, we can find that the Z-score for a right-tail probability of 0.0062 is approximately -2.56.
Substituting the values into the Z-score formula, we can solve for X:
-2.56 = (65 - x) / 8
Multiply both sides by 8:
-2.56 * 8 = 65 - x
-20.48 = 65 - x
Rearranging the equation, we find:
x = 65 - 20.48
x ≈ 44.52
Therefore, the value of x, which represents the mean installation time for the new telephone, is approximately 44.52 minutes.