To calculate the probability that cell phone usage will be above 5 hours per day, we first need to find the sample mean and standard deviation from the 30 data points. Then, we can use this information to estimate the probability.

Steps:

1. Calculate the sample mean.
2. Calculate the sample standard deviation.
3. Standardize the value 5 hours using the z-score formula.
4. Find the probability associated with that z-score.

1. Sample Mean

We are given the following data points:

$\text{Data points} = [5.34, 6.17, 5.24, 1.35, 3.53, 4.63, 1.28, 2.41, 6.18, 5.29, 3.59, 2.19, 3.14, 7.74, 2.96, 0.36, 1.78, 0.79, 5.26, 3.21, 2.07, 6.20, 4.81, 4.82, 7.02, 6.45, 6.52, 7.74, 7.40, 5.00]$

The sample mean $\mu$ is calculated as:

$\mu = \frac{\sum X}{n}$

Where:

• $X$ is each individual data point.
• $n$ is the number of data points (in this case, $n = 30$).

2. Sample Standard Deviation

The sample standard deviation $s$ is calculated using the formula:

$s = \sqrt{\frac{\sum (X_i - \mu)^2}{n - 1}}$

Where $X_i$ represents each data point, and $\mu$ is the sample mean.

3. Z-score for 5 Hours

To determine the probability of having more than 5 hours of usage, we will calculate the z-score for 5 hours:

$z = \frac{X - \mu}{s}$

Where:

• $X = 5$ (the value we are interested in),
• $\mu$ is the sample mean,
• $s$ is the sample standard deviation.

4. Probability Calculation

Once we calculate the z-score, we can use the standard normal distribution table to find the probability that a random data point exceeds 5 hours of usage. This is done by determining the area to the right of the z-score.

Let me perform the calculations now. numpy as np import scipy.stats as stats

Data points

data = [ 5.34, 6.17, 5.24, 1.35, 3.53, 4.63, 1.28, 2.41, 6.18, 5.29, 3.59, 2.19, 3.14, 7.74, 2.96, 0.36, 1.78, 0.79, 5.26, 3.21, 2.07, 6.20, 4.81, 4.82, 7.02, 6.45, 6.52, 7.74, 7.40, 5.00 ]

Step 1: Calculate sample mean

mean = np.mean(data)

Step 2: Calculate sample standard deviation

std_dev = np.std(data, ddof=1)

Step 3: Z-score for 5 hours

X = 5 # Value we are interested in z_score = (X - mean) / std_dev

Step 4: Find the probability (area to the right of the z-score)

probability = 1 - stats.norm.cdf(z_score)

mean, std_dev, z_score, probabilityHere are the results of the calculations:

1. Sample mean: 4.349 hours
2. Sample standard deviation: 2.167 hours
3. Z-score for 5 hours: 0.300
4. Probability of cell phone usage being greater than 5 hours: 0.382, or approximately 38.2%.

This means there is about a 38.2% chance that a randomly selected individual in this sample uses their phone for more than 5 hours a day.

Would you like to dive deeper into any of these steps, or do you have any other questions? Here's a few related questions to explore:

1. How does the z-score relate to the normal distribution curve?
2. What does the sample standard deviation tell us about the data?
3. How would the probability change if we considered the population mean instead of the sample mean?
4. How can we interpret the results if the sample size was smaller or larger?
5. What assumptions are made when using the normal distribution for this probability calculation?

Tip: When calculating probabilities using the z-score, make sure to check whether your data approximates a normal distribution. The larger the sample size, the more likely the Central Limit Theorem ensures normality.