To construct a 99% confidence interval for the population mean screen time, we follow these steps:

Step 1: Identify the Given Data

We are given a dataset of 48 screen times (already rounded to the nearest tenth of an hour). To compute the confidence interval, we need:

• Sample size (n) = 48
• Sample mean (x̄) = ?
• Sample standard deviation (s) = ?
• Confidence level = 99% (which corresponds to a critical value, $t^*$, from the t-distribution)

Since the population standard deviation ($\sigma$) is unknown, we use the t-distribution with degrees of freedom (df = n - 1 = 47).

Step 2: Compute the Sample Mean ($x̄$) and Sample Standard Deviation ($s$)

Let's calculate these values. numpy as np

Given screen time data

data = np.array([ 3.4, 4.2, 5.1, 4.8, 9.4, 7.2, 6.5, 4.5, 5.2, 5.9, 6.3, 7.2, 9.5, 8.3, 4.4, 7.6, 10.2, 8.6, 2.3, 6.5, 4.3, 6.2, 5.7, 7.2, 7.8, 8.9, 2.4, 7.6, 6.3, 3.9, 4.3, 5.2, 8.4, 6.1, 5.7, 6.3, 6.1, 5.8, 5.3, 4.2, 4.8, 6.4, 6.8, 8.3, 4.1, 5.5, 3.7 ])

Calculate mean and standard deviation

sample_mean = np.mean(data) sample_std = np.std(data, ddof=1) # Using ddof=1 for sample standard deviation sample_size = len(data)

sample_mean, sample_std, sample_sizeWe have calculated:

• Sample Mean ($x̄$) = 6.0511 hours
• Sample Standard Deviation ($s$) = 1.8467 hours
• Sample Size ($n$) = 47

Now, we find the critical value $t^*$ for a 99% confidence level with df = 47. scipy.stats as stats

Find the critical t-value for 99% confidence interval with df = 47

confidence_level = 0.99 df = sample_size - 1 # Degrees of freedom

t_critical = stats.t.ppf((1 + confidence_level) / 2, df) t_criticalThe critical t-value ($t^*$) for a 99% confidence level with 47 degrees of freedom is 2.6870.

Now, we compute the margin of error (ME) using the formula:

$ME = t^* \times \frac{s}{\sqrt{n}}$

Then, we calculate the confidence interval:

$\text{Lower Bound} = x̄ - ME$

$\text{Upper Bound} = x̄ + ME$ Compute margin of error margin_of_error = t_critical * (sample_std / np.sqrt(sample_size))

Compute confidence interval

lower_bound = sample_mean - margin_of_error upper_bound = sample_mean + margin_of_error

margin_of_error, lower_bound, upper_boundThe 99% confidence interval for the population mean screen time is:

$(5.3273, 6.7749)$

Final Answer:

$\mathbf{(5.33, 6.77)}$ (rounded to two decimal places)

This means we are 99% confident that the true mean screen time of the population falls within this interval.

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Would you like a breakdown of any step? Here are some related questions you might be interested in:

1. What is the interpretation of a confidence interval in statistics?
2. How does increasing the confidence level affect the confidence interval width?
3. When should we use the t-distribution instead of the normal distribution?
4. How do outliers impact confidence interval estimation?
5. What assumptions must be met for a t-interval to be valid?

Tip: Always check if the sample size is sufficiently large for the Central Limit Theorem to apply when estimating population parameters!