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.