To determine the percentage of teenagers who spend more than 3.1 hours per day on a brand A cell phone, we need to perform the following steps:
### Step 1: Define the Given Values
- Mean daily time, [tex]\(\mu = 2.5 \, \text{hr}\)[/tex]
- Standard deviation, [tex]\(\sigma = 0.6 \, \text{hr}\)[/tex]
- Threshold time, [tex]\(3.1 \, \text{hr}\)[/tex]
### Step 2: Calculate the Z-Score
The Z-score formula for a given value [tex]\(X\)[/tex] is:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
Using the given threshold [tex]\(3.1 \, \text{hr}\)[/tex]:
[tex]\[ Z = \frac{3.1 - 2.5}{0.6} = \frac{0.6}{0.6} = 1.0 \][/tex]
### Step 3: Determine the Probability Corresponding to the Z-Score
A Z-score of 1.0 corresponds to the cumulative probability from the standard normal distribution table.
[tex]\[ P(X \leq 3.1) = \text{CDF}(1.0) \][/tex]
From standard normal distribution tables or computational tools, the cumulative distribution function (CDF) value at [tex]\(Z = 1.0\)[/tex] is approximately [tex]\(0.8413\)[/tex]. This value represents the probability that a teenager spends 3.1 hours or less per day on the phone.
### Step 4: Calculate the Complementary Probability
To find the probability that a teenager spends more than 3.1 hours per day, we calculate the complement of the CDF value:
[tex]\[ P(X > 3.1) = 1 - P(X \leq 3.1) \][/tex]
[tex]\[ P(X > 3.1) = 1 - 0.8413 = 0.1587\][/tex]
### Step 5: Convert Probability to Percentage
Finally, to express this probability as a percentage:
[tex]\[ \text{Percentage} = 0.1587 \times 100 = 15.87\% \][/tex]
Thus, approximately 15.87% of teenagers spend more than 3.1 hours per day on a brand A cell phone. The closest option from the provided choices is:
[tex]\[ \boxed{16\%} \][/tex]
