A study indicates that​ 18 to​ 24 year olds spend a mean of 115 minutes watching video on their smartphones per month. Assume that the amount of time watching video on a smartphone per month is normally distributed and that the standard deviation is 20 minutes. What is the probability that an​ 18 to​ 24 year-old spends less than 85 minutes watching video on his or her smartphone per​ month?

To find the probability that an 18 to 24-year-old spends less than 85 minutes watching video on their smartphone per month, we can use the properties of the normal distribution.

Given:
- Mean (\(\mu\)) = 115 minutes
- Standard deviation (\(\sigma\)) = 20 minutes
- We want to find \(P(X < 85)\), where \(X\) is the amount of time spent watching videos.

### Steps to solve:

1. **Calculate the z-score** for 85 minutes using the formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   Substituting the values:
   \[
   z = \frac{85 - 115}{20} = \frac{-30}{20} = -1.5
   \]

2. **Find the probability** corresponding to the z-score of \(-1.5\).
   - Using a z-table or a statistical calculator, the probability corresponding to a z-score of \(-1.5\) is approximately 0.0668.

### Conclusion:

The probability that an 18 to 24-year-old spends less than 85 minutes watching video on their smartphone per month is **0.0668**, or **6.68%**.

Would you like any further explanation on this, or have any other questions?

Here are five related questions that might interest you:
1. How would the probability change if the standard deviation were larger?
2. What is the probability that an 18 to 24-year-old spends more than 130 minutes watching videos?
3. How can we find the time threshold above which only 10% of the 18 to 24-year-olds spend watching videos?
4. How would you interpret a z-score in terms of standard deviations from the mean?
5. What is the significance of the area under the normal curve?

**Tip:** When calculating probabilities from a normal distribution, the z-score helps translate raw data into a standard format, making it easier to find probabilities and make comparisons.

Calculate the probability that an 18 to 24-year-old spends less than 85 minutes watching video on their smartphone per month using the normal distribution and z-score. Understand the properties of the normal distribution, mean, and standard deviation involved in statistical calculations.

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