Given the prompt, let's go through the detailed step-by-step solution:
1. Understand the Parameters
- Mean, [tex]\(\mu = 2.5\)[/tex] hours
- Standard deviation, [tex]\(\sigma = 0.6\)[/tex] hours
- The time of interest, [tex]\(x = 3.1\)[/tex] hours
2. Calculate the Z-Score
The Z-score tells us how many standard deviations an element is from the mean.
[tex]\[
Z = \frac{(X - \mu)}{\sigma}
\][/tex]
Plug in the given numbers:
[tex]\[
Z = \frac{(3.1 - 2.5)}{0.6} = \frac{0.6}{0.6} = 1.0
\][/tex]
3. Interpret the Z-Score
A Z-score of 1.0 means that 3.1 hours is one standard deviation above the mean.
4. Find the Cumulative Probability
The cumulative probability (area under the normal curve to the left of Z) can be found from standard normal distribution tables or a standard normal distribution calculator, which gives:
[tex]\[
P(Z \leq 1.0) \approx 0.8413 \quad (or \, 84.13\%)
\][/tex]
5. Calculate the Percentage Spending More than 3.1 Hours
To find the percentage of teenagers spending more than 3.1 hours, we need to find the complementary probability (area to the right of Z).
[tex]\[
P(X > 3.1) = 1 - P(Z \le 1.0) = 1 - 0.8413 = 0.1587 \quad (or \, 15.87\%)
\][/tex]
Thus, approximately 15.87% of teenagers spend more than 3.1 hours on a brand A cell phone daily.
Checking against the provided options:
- 5%
- 10%
- 16%
- 32%
The closest match to our calculated percentage (15.87%) is 16%. Therefore, the answer is:
[tex]\[
\boxed{16\%}
\][/tex]
