The daily screen time of students at the University of Michigan follows an approximate normal distribution. 

Two students, Justin and Nasir, provided their screen times.  Justin reports he typically spends about 245 minutes on his phone each day, a screen time that places him in the top 5% of students. Nasir reports typically using his phone 98 minutes a day, which places him in the bottom 15% of students.

Given this information, what is the mean amount of time (in minutes) students at the University of Michigan typically spend on their phone each day? Do not use a z score chart

Given that the daily screen time of students follows an approximate normal distribution, we need to determine the mean \( \mu \) of the distribution using the information provided for two students, Justin and Nasir, without using a z-score chart. 

### Step-by-step breakdown:

1. **Justin’s Screen Time (Top 5%)**:
   - Justin spends 245 minutes on his phone each day, and this places him in the top 5% of the distribution.
   - In a normal distribution, the top 5% corresponds to a z-score of approximately \( z = 1.645 \) because 95% of the data falls below this value.
   - This means that Justin's screen time corresponds to a z-score of \( z_J = 1.645 \).
   
2. **Nasir’s Screen Time (Bottom 15%)**:
   - Nasir spends 98 minutes on his phone each day, placing him in the bottom 15% of the distribution.
   - For the bottom 15%, the corresponding z-score is approximately \( z = -1.036 \) because the z-score that leaves 15% of the data to the left is \( -1.036 \).
   - This means Nasir's screen time corresponds to a z-score of \( z_N = -1.036 \).

### Using the z-score formula:
The z-score formula is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \( x \) is the observed value (screen time),
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation of the distribution.

Using this formula, we can set up two equations based on the given information:

1. For Justin:
\[
1.645 = \frac{245 - \mu}{\sigma}
\]
(Equation 1)

2. For Nasir:
\[
-1.036 = \frac{98 - \mu}{\sigma}
\]
(Equation 2)

### Solving the system of equations:
Now, we solve these two equations simultaneously.

#### Step 1: Solve each equation for \( \sigma \):

From Equation 1:
\[
\sigma = \frac{245 - \mu}{1.645}
\]

From Equation 2:
\[
\sigma = \frac{98 - \mu}{-1.036}
\]

#### Step 2: Set the two expressions for \( \sigma \) equal to each other:
\[
\frac{245 - \mu}{1.645} = \frac{98 - \mu}{-1.036}
\]

#### Step 3: Cross-multiply and solve for \( \mu \):
\[
(245 - \mu)(-1.036) = (98 - \mu)(1.645)
\]

Expanding both sides:
\[
-1.036 \times 245 + 1.036 \mu = 1.645 \times 98 - 1.645 \mu
\]
\[
-253.82 + 1.036 \mu = 161.21 - 1.645 \mu
\]

#### Step 4: Combine like terms:
\[
1.036 \mu + 1.645 \mu = 161.21 + 253.82
\]
\[
2.681 \mu = 415.03
\]

#### Step 5: Solve for \( \mu \):
\[
\mu = \frac{415.03}{2.681} \approx 154.81
\]

Thus, the mean screen time \( \mu \) for students at the University of Michigan is approximately **154.81 minutes** per day.

Would you like any more details or explanations on this solution?

### Related questions:
1. How can you calculate the standard deviation of the screen time distribution using the provided information?
2. If another student reports spending 200 minutes a day on their phone, what percentile would they be in?
3. How does the z-score value relate to percentiles in a normal distribution?
4. What are the characteristics of a normal distribution that make problems like this solvable?
5. How would the problem change if the distribution of screen times was not normal?

### Tip:
Understanding the relationship between z-scores and percentiles in a normal distribution allows for easier interpretation of data points within any distribution.

The daily screen time of students at the University of Michigan follows an approximate normal distribution. Two students, Justin and Nasir, provided their screen times. Justin reports he typically spends about 245 minutes on his phone each day, a screen time that places him in the top 5% of students. Nasir reports typically using his phone 98 minutes a day, which places him in the bottom 15% of students. Given this information, what is the mean amount of time (in minutes) students at the University of Michigan typically spend on their phone each day? Do not use a z score chart.

Solve a problem involving the mean phone screen time of University of Michigan students using normal distribution. Given data from two students with screen times in the top 5% and bottom 15%, learn how to calculate the average screen time with step-by-step explanations. Suitable for college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation