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The problem is based on a normally distributed random variable, where the mean daily screen time is \( \mu = 8.35 \) hours and the standard deviation is \( \sigma = 2.5 \) hours. Let's break down each part of the question:

### Part (a) - Probability between 5 and 9 hours of screen time
You are asked to find the probability that a person has between 5 and 9 hours of screen time. We need to convert these values to **Z-scores** and then use the standard normal distribution table (or a calculator).

The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]
For 5 hours:
\[
Z_5 = \frac{5 - 8.35}{2.5} = \frac{-3.35}{2.5} = -1.34
\]
For 9 hours:
\[
Z_9 = \frac{9 - 8.35}{2.5} = \frac{0.65}{2.5} = 0.26
\]

Now, looking up these Z-scores in a standard normal table or using a calculator:
- \( P(Z = -1.34) \approx 0.0901 \)
- \( P(Z = 0.26) \approx 0.6026 \)

So, the probability that screen time is between 5 and 9 hours is:
\[
P(5 < X < 9) = P(0.26) - P(-1.34) = 0.6026 - 0.0901 = 0.5125
\]

### Part (b) - Top 6% of screen time
Here, we need to find the value of screen time that corresponds to the top 6% of the distribution. This means we are looking for the 94th percentile (since 100% - 6% = 94%).

Using a standard normal distribution table, the Z-score corresponding to the 94th percentile is approximately \( Z = 1.555 \).

Now, convert the Z-score back to the original value using the formula:
\[
X = \mu + Z \sigma = 8.35 + 1.555 \times 2.5 = 8.35 + 3.89 = 17.00 \text{ hours}
\]

### Part (c) - Probability of more than 3 hours of screen time
We need to find the probability that a person has more than 3 hours of screen time.

First, convert 3 hours to a Z-score:
\[
Z_3 = \frac{3 - 8.35}{2.5} = \frac{-5.35}{2.5} = -2.14
\]

Now, look up the Z-score:
- \( P(Z = -2.14) \approx 0.0162 \)

So, the probability of more than 3 hours of screen time is:
\[
P(X > 3) = 1 - P(Z = -2.14) = 1 - 0.0162 = 0.9838
\]

---

Would you like more details or have any questions?

### Related Questions:
1. What is the Z-score for a person with 12 hours of screen time?
2. How would the results change if the standard deviation were 3 hours instead of 2.5 hours?
3. What is the probability that a person has less than 8 hours of screen time?
4. How many hours correspond to the top 10% of screen time values?
5. What is the probability that a person has between 6 and 10 hours of screen time?

### Tip:
Always remember to convert raw values to Z-scores when dealing with normal distributions to standardize your calculations.

Suppose that the mean time someone spends using a device with a screen such as a smartphone, computer, or television is 8.35 hours. Use a normal probability distribution with a standard deviation of 2.5 hours to answer the following questions about daily screen time.

a. What is the probability that a randomly selected person has between 5 and 9 hours of screen time a day (to 4 decimals)?

b. How many hours of screen time must a person have in order to be in the top 6% of all screen time values (to 2 decimals)?

c. What is the probability that a person has more than 3 hours of screen time a day (to 4 decimals)?

Explore screen time statistics using a normal distribution with a mean of 8.35 hours and a standard deviation of 2.5 hours. Learn how to calculate probabilities between given time ranges, find the top 6% of screen time values, and determine the probability of more than 3 hours of daily screen time. Suitable for college-level statistics students.

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