Questions: A few cell phones of middle school children were evaluated for hours used. The mean time is 25.4 hours per month and the standard deviation 0.4 hours. How many cell phones must be sampled to be 95% confident that the sample mean does not differ from the population mean within ± 0.2 hours?

Solution Steps

We are provided with the following parameters:

• Population mean (\( \mu \)) = 25.4 hours
• Population standard deviation (\( \sigma \)) = 0.4 hours
• Desired confidence level = 95%
• Margin of error (\( E \)) = 0.2 hours

For a 95% confidence level, the Z-score (\( Z \)) is approximately: \[ Z \approx 1.96 \]

The formula for calculating the required sample size (\( n \)) is given by: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] Substituting the known values: \[ n = \left( \frac{1.96 \cdot 0.4}{0.2} \right)^2 \] Calculating the expression: \[ n = \left( \frac{0.784}{0.2} \right)^2 = (3.92)^2 = 15.3664 \] Since the sample size must be a whole number, we round up to the nearest whole number: \[ n = 16 \]

Final Answer