To calculate the 95% confidence interval for the average screen time, we will use the formula for the confidence interval for the population mean when the population standard deviation is known. The formula is given by:
\[ \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \]
Where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score corresponding to the desired confidence level
- \(\sigma\) = population standard deviation
- \(n\) = sample size
Given:
- Sample mean (\(\bar{x}\)) = 53 minutes
- Population standard deviation (\(\sigma\)) = 13 minutes
- Sample size (\(n\)) = 80
Step 1: Determine the z-score for the 95% confidence level. For a 95% confidence level, the z-score (z) is approximately 1.96.
Step 2: Calculate the standard error (SE). \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{13}{\sqrt{80}} \approx \frac{13}{8.9443} \approx 1.455 \]
Step 3: Calculate the margin of error (ME). \[ ME = z \times SE = 1.96 \times 1.455 \approx 2.855 \]
Step 4: Calculate the confidence interval. \[ \text{Confidence Interval} = \bar{x} \pm ME = 53 \pm 2.855 \]
Lower limit: \[ 53 - 2.855 \approx 50.145 \]
Upper limit: \[ 53 + 2.855 \approx 55.855 \]
Final 95% Confidence Interval: \[ (50.15, 55.86) \]
Thus, the 95% confidence interval for the average screen time is approximately \((50.15, 55.86)\) minutes.