Let X denote the time taken for a computer link to be made between the terminal in an executive's office and the computer at a remote factory site.  It is known that X has a Normal distribution with a mean of 15 seconds and a standard deviation of 3 seconds.  On 90% of the occasions the computer link is made in less than

We are given that the time \( X \) to establish the computer link follows a normal distribution with:

- Mean \( \mu = 15 \) seconds,
- Standard deviation \( \sigma = 3 \) seconds.

We are tasked with finding the time \( T \) such that the computer link is made in less than \( T \) seconds on 90% of the occasions. This is equivalent to finding \( T \) such that:

\[
P(X < T) = 0.90
\]

This can be solved by standardizing the variable \( X \) to a standard normal distribution (with mean 0 and standard deviation 1) using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

We want to find the value of \( Z \) such that:

\[
P(Z < z) = 0.90
\]

From standard normal distribution tables, the z-score corresponding to a cumulative probability of 0.90 is approximately \( z = 1.28 \).

Now, we convert this z-score back to the original scale:

\[
T = \mu + z \cdot \sigma = 15 + 1.28 \cdot 3
\]

\[
T = 15 + 3.84 = 18.84 \text{ seconds}
\]

Thus, on 90% of the occasions, the computer link is made in less than approximately **18.84 seconds**.

Would you like further details or have any questions about this?

Here are 5 related questions for further practice:
1. What is the probability that the computer link takes more than 20 seconds?
2. If the mean time changes to 17 seconds, what is the new time at which 90% of links are made?
3. What percentage of links are made in less than 12 seconds?
4. If the standard deviation increases to 4 seconds, what would be the time corresponding to the 90th percentile?
5. What is the probability that the link time is between 12 and 18 seconds?

**Tip:** The z-score table helps you convert between raw scores and probabilities in a normal distribution.

Let X denote the time taken for a computer link to be made between the terminal in an executive's office and the computer at a remote factory site. It is known that X has a Normal distribution with a mean of 15 seconds and a standard deviation of 3 seconds. On 90% of the occasions the computer link is made in less than...

Solve a probability problem involving a normal distribution with a mean of 15 seconds and a standard deviation of 3 seconds. Find the time corresponding to the 90th percentile. Learn key concepts such as the z-score formula and cumulative probability.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation